A technique used to solve systems of linear equations by eliminating one variable at a time.
Vector addition involves combining two vectors to produce a third vector.
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others.
Diagonalizing a matrix involves transforming it into a diagonal form, which simplifies many matrix operations and computations.
Diagonal matrices are square matrices in which all the entries outside the main diagonal are zero.
A free variable in eigenvalue problems indicates that it can take any value, leading to infinitely many solutions for the corresponding eigenvector.
The matrix representation of a linear transformation is a matrix that describes how the transformation acts on vectors in the vector space, relative to specified bases.
The Span of a set of vectors is the set of all possible linear combinations of those vectors.
Linear transformations are just matrices in fancy wrapping.
Linearity ensures that the transformation maintains the structure of the vector space, meaning that the output is a linear combination of the inputs.
A matrix equation is an equation in which matrices are used to represent a system of linear equations, typically in the form Ax = b.
The three properties are: 1) the zero vector is in the subset, 2) the subset is closed under vector addition, and 3) the subset is closed under scalar multiplication.
A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of the others.
A basis for a vector space is a set of vectors that is linearly independent and spans the vector space.
A vector is in the Span if and only if the equation formed by the vectors has a solution.
A basis for a vector space is a set of vectors that are linearly independent and span the space.
It means that the transformation can be uniquely defined by the images of the basis vectors.
The vector equation is used to determine if the set of vectors can be expressed as a linear combination of each other.
Vectors are linearly independent if no vector in the set can be expressed as a linear combination of the others.
The diagonal entries of a triangular matrix are the eigenvalues.
To prove a property of a linear transformation means to demonstrate that the transformation satisfies certain conditions or equations for all vectors in the vector space.
A variable in a system of equations that can take on any value, leading to multiple solutions.
The linear span of a set of vectors is the set of all possible linear combinations of those vectors.
A subspace of a vector space is a subset that is itself a vector space under the same operations of addition and scalar multiplication defined on the larger vector space.
A square matrix is diagonalizable if and only if there are linearly independent eigenvectors. Additionally, if a matrix has distinct eigenvalues, it must be diagonalizable.
The subset of all linear combinations of a set of vectors is called the span of those vectors.
A method to solve two or more equations at the same time to find the values of the variables that satisfy all equations.
Vectors are linearly independent if no vector in the set can be expressed as a scalar multiple of another vector in the set.
The line through the origin that contains a non-zero vector is defined as the set of all scalar multiples of that vector.
A change-of-coordinates matrix is a matrix that transforms the coordinates of a vector from one basis to another in a vector space.
A system is consistent if it has at least one solution.
A linear combination of vectors is defined by a combination of given vectors and scalars, resulting in a new vector.
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
The solution set represents all possible solutions that satisfy the system of linear equations.
It means that there exist values for the unknowns that satisfy the equation, indicating that the vector can be expressed as a linear combination of other vectors.
The dimension of a vector space is defined to be the number of vectors in any basis for that vector space.
The standard basis for R^n consists of vectors that have a 1 in one coordinate and 0 in all others, serving as the building blocks for the space.
Computing the matrix in the example demonstrates the practical application of diagonalization in simplifying matrix operations.
A basis is a set of vectors in a vector space that is linearly independent and spans the entire space.
A change-of-coordinates matrix is a matrix that transforms vectors from one basis to another basis in a vector space.
A vector equation is an equation that expresses a relationship between vectors, typically involving a linear combination of vectors set equal to another vector.
A collection of vectors in a vector space is said to be linearly independent if the vector equation has ONLY the trivial solution.
An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations.
Given a linear transformation and a basis for a vector space, if certain elements are fixed, then the transformation is completely determined.
A coefficient matrix is a matrix that contains the coefficients of the variables in a system of linear equations.
The elimination method is used to solve systems of linear equations by eliminating variables to simplify the equations into a form that can be easily solved.
A function that does not satisfy the properties of linearity, specifically failing to meet Property 1 in the definition of linear functions.
A basis for a vector space is a set of vectors that are linearly independent and span the entire space.
Property 1 states that for a function to be linear, it must satisfy the condition f(x + y) = f(x) + f(y) for all vectors x and y.
The change-of-coordinates matrix is a matrix that transforms coordinates of vectors from one basis to another in a vector space.
A vector space is a set of elements (such as vectors) that can be added together and multiplied by scalars, satisfying certain axioms.
The characteristic polynomial is a polynomial which is derived from the determinant of a matrix subtracted by a scalar multiple of the identity matrix, used to find eigenvalues.
A collection of vectors in a vector space is a basis if it is a linearly independent set and the vectors span the space.
A coordinate vector is a representation of a vector in terms of the basis vectors of a vector space.
The -coordinates of a vector are the scalars that express the vector as a linear combination of the basis vectors.
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
An augmented matrix is a matrix that includes the coefficients of a linear system along with the constants from the equations, used to solve the system.
Diagonalizing a matrix involves finding a diagonal matrix that is similar to the original matrix, which means there exists an invertible matrix such that the original matrix can be expressed as the product of the invertible matrix, the diagonal matrix, and the inverse of the invertible matrix.
The dimension of the subspace is determined by the number of linearly independent vectors in the set, which in this case is 3.
A vector is a mathematical object that has both a magnitude and a direction, often represented as an ordered pair or triplet of numbers in a coordinate system.
A linear system is inconsistent if there are no solutions that satisfy all equations in the system.
A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.
A solution to a linear system is a set of values for the variables that satisfies all equations in the system.
It indicates that the two vectors are linearly dependent, meaning they do not span a space larger than one dimension.
An eigenvalue is a scalar associated with a linear transformation represented by a matrix, indicating how much a corresponding eigenvector is stretched or compressed during the transformation.
Matrices that have a staircase-like structure where each leading entry of a row is to the right of the leading entry of the previous row.
A matrix is in reduced echelon form if it satisfies the following conditions: each leading entry is 1, each leading 1 is the only non-zero entry in its column, and the leading 1s move to the right as you move down the rows.
A non-trivial solution is a solution to an equation that is not the zero solution, meaning at least one variable has a non-zero value.
Matrices that are in echelon form with the additional property that each leading entry is 1 and is the only non-zero entry in its column.
A matrix equation is an equation in which the unknowns are represented by matrices, and it can often be expressed in terms of vector equations.
It means that the sum of any two vectors in the set is also a vector in the set.
A vector in the context of bases is an element of a vector space that can be expressed as a linear combination of the basis vectors.
A basis is linearly independent if no vector in the basis can be expressed as a linear combination of the other vectors in the basis.
An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector.
A free variable is a variable in a linear system that can take on any value, indicating that the system has infinitely many solutions due to the number of unknowns being greater than the number of equations.
The set of all eigenvectors corresponding to an eigenvalue λ, together with the zero vector, is called the eigenspace of T corresponding to λ.
Finding a vector in a given basis involves expressing the vector as a linear combination of the basis vectors.
A subspace is a subset of a vector space that is itself a vector space, satisfying three properties: it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication.
Coordinate Systems are frameworks that use numbers to uniquely determine the position of a point or other geometric element in a space of given dimensions.
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
Having two vectors refers to the existence of two distinct quantities that have both magnitude and direction, which can be represented graphically or mathematically.
A linear combination is an expression formed by multiplying each vector in a set by a scalar and then adding the results together.
A set of only two vectors is linearly dependent if and only if one is a scalar multiple of the other.
A clever method or technique used to simplify the process of solving equations or problems.
If two bases for a vector space contain the same number of vectors, it implies that the dimension of the vector space is well-defined.
The characteristic equation is a polynomial equation derived from the determinant of a matrix subtracted by a scalar multiple of the identity matrix, used to find the eigenvalues of the matrix.
An augmented matrix is a matrix that includes the coefficients of a system of linear equations along with the constants from the equations, typically represented in a single matrix format.
A basis for a vector space is a set of vectors that are linearly independent and span the entire vector space.
Two bases of a vector space are related if there exists a linear transformation that maps vectors from one basis to the other.
A vector is in a vector space if and only if the vector equation associated with that space has a solution.
The set of all vectors with three entries is denoted as R^3.
An eigenvector is a non-zero vector that changes by only a scalar factor when a linear transformation is applied to it.
Every matrix can be changed to a unique reduced echelon matrix by row operations.
Diagonalization is the process of finding a diagonal matrix that is similar to a given square matrix, which simplifies many matrix operations.
A matrix with only one column is called a column vector, or simply a vector.
In this step, you implement the actions identified in the previous step to begin solving the problem.
A vector equation expresses a linear system in terms of vectors, showing the relationship between the variables and the constants in vector form.
A basis is a set of vectors in a vector space that is linearly independent and spans the entire space.
Eigenvalues are scalars associated with a linear transformation that, when multiplied by an eigenvector, yield the same result as applying the transformation to that eigenvector.
The change-of-coordinates matrix is a matrix that transforms the coordinates of a vector from one coordinate system to another, specifically to the standard coordinates.
The Geometry of Vectors involves representing vectors graphically, showing their direction and magnitude in a coordinate system.
A basis for a subspace is a set of vectors that are linearly independent and span the subspace.
The determinant of a triangular matrix (upper or lower) equals the product of its diagonal entries.
Eigenvalues are scalars associated with a linear transformation represented by a matrix, indicating how much the eigenvector is stretched or compressed during the transformation.
Diagonalizing a matrix involves finding a diagonal matrix that is similar to the original matrix, which simplifies many matrix operations.
A vector is in a subspace if it satisfies the conditions of closure under addition and scalar multiplication within that subspace.
Solve the resulting equation for the remaining variable and back-substitute to find the other variable.
Each column vector of the matrix can be viewed as the coordinate vector of itself in a specific basis.
It indicates that the system yields non-trivial solutions, suggesting that the vector equation is linearly dependent.
This step involves analyzing the problem and determining the necessary actions to solve it.
An eigenvector is a non-zero vector that changes by only a scalar factor when a linear transformation is applied to it.
The non-zero vector v that satisfies the equation T(v) = λv for an eigenvalue λ is called an eigenvector corresponding to λ.
R^n is the set of all vectors with n entries, where n is any positive integer.
The conditions are that the set must be linearly independent and the vectors must span the vector space.
The two operations defined in a vector space are addition and scalar multiplication.
To prove that a function is a linear transformation, one must show that it satisfies two properties: it preserves vector addition and scalar multiplication.
A square matrix is said to be diagonalizable if there exists a diagonal matrix and an invertible matrix such that the matrix equation is satisfied.
The characteristic equation is obtained by setting the characteristic polynomial equal to zero, used to solve for the eigenvalues of a matrix.
This step focuses on evaluating the results of the actions taken to see if the problem has been solved.
A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero, and the entries on the diagonal can be any value.
An example of an inconsistent linear system is one where the equations represent parallel lines that never intersect, indicating no common solution.
A scalar is a single numerical value that can be used to scale a vector, affecting its magnitude but not its direction.
The property that allows two systems of equations to have the same solution set, often verified through manipulation.
The change-of-coordinates matrix is a matrix that transforms the coordinate vector of a vector in one basis to its coordinate vector in another basis within a vector space.
The plane through the origin that contains two non-zero vectors is defined as the set of all linear combinations of those two vectors.
Bases in a vector space are sets of linearly independent vectors that span the entire space, allowing any vector in the space to be expressed as a linear combination of the basis vectors.
A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
A scalar λ is called an eigenvalue of a linear transformation T if there exists a non-zero vector v such that T(v) = λv.
The span of a set of vectors is the set of all possible linear combinations of those vectors.
A vector space is a non-empty set on which two operations, called addition and scalar multiplication, are defined that satisfy specific properties.
Non-trivial solutions refer to solutions other than the zero vector, indicating that the vectors involved are linearly dependent.
A transformation (or function or mapping) from one vector space to another is a rule that assigns to each vector in the domain a vector in the range.
An eigenvalue is a scalar associated with a linear transformation that indicates how much a corresponding eigenvector is stretched or compressed.
Real coefficients ensure that the polynomials belong to the set of real-valued functions, which is necessary for defining a vector space.
It means that multiplying any vector in the set by a scalar results in another vector that is also in the set.
A coordinate system allows one to represent the vectors in a vector space in a manner similar to vectors in Euclidean space, facilitating operations and understanding of the space.
The smallest subspace of a vector space is the set that consists of only the zero vector.
The domain is the set of all vectors that the transformation takes as input.
An eigenvalue is a scalar associated with a linear transformation that indicates how much a corresponding eigenvector is stretched or compressed.
The characteristic equation is the equation derived from the determinant of a matrix minus a scalar times the identity matrix, which is used to find the eigenvalues of the matrix.
The zero vector is an element in a vector space such that for each vector in the space, there exists another vector that, when added to it, results in the zero vector.
Non-trivial solutions refer to solutions of a linear system that are not the zero solution, indicating that the system has free variables and is linearly dependent.
A linear transformation from vector space A to vector space B is a transformation that satisfies the properties of additivity and homogeneity for any vectors u and v in A and any scalar c.
The reduced echelon form is a specific type of matrix form where each leading entry is 1, each leading 1 is the only non-zero entry in its column, and the leading 1s move to the right as you move down the rows.
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others.
Vectors are linearly dependent if there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector.
If a matrix is obtained by performing row (or column) operations on another matrix, the determinant may change according to specific rules related to the type of operation performed.
The Change of Basis theorem states that for two bases of a vector space, there exists a matrix that changes the coordinate representation of vectors from one basis to another.
Identify and manipulate the equations to eliminate one variable.
Vectors are linearly independent if no vector in the set can be expressed as a linear combination of the others.
A subspace is a set of vectors that is closed under vector addition and scalar multiplication, containing the zero vector.
A matrix equation is a mathematical expression that represents a system of linear equations in the form of a product of matrices.
A coordinate vector represents a vector in terms of the basis vectors of a vector space.
A vector equation is an equation that expresses a relationship between vectors, often representing a system of linear equations in a compact form.
An eigenvalue is a scalar associated with a linear transformation that indicates how much the corresponding eigenvector is stretched or compressed.
Eigenvalues are scalars associated with a linear transformation represented by a matrix, indicating how much the eigenvector is stretched or compressed during the transformation.
Vectors are linearly dependent if at least one vector can be expressed as a linear combination of the others, leading to non-trivial solutions.
The set of all real numbers is a fundamental concept in mathematics that includes all the numbers on the continuous number line.
The two properties are: 1) For any vectors u and v in the vector space, T(u + v) = T(u) + T(v); 2) For any scalar c and vector u, T(cu) = cT(u).
An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector.
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others.
A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
A subspace of a vector space is a subset that satisfies three properties: it contains the zero vector, it is closed under addition, and it is closed under scalar multiplication.
A vector space is a collection of vectors that can be added together and multiplied by scalars, satisfying certain axioms such as closure, associativity, and distributivity.
The largest subspace of a vector space is the vector space itself.
An eigenspace is the set of all eigenvectors associated with a particular eigenvalue, along with the zero vector.
An augmented matrix is a matrix that includes the coefficients of a system of linear equations along with the constants from the equations, typically used in the elimination method.
The matrix representation of a linear transformation depends on the chosen bases for the domain and codomain.
A vector is said to be in a space if it can be expressed as a linear combination of the vectors that span that space.
The coordinates of a vector in R^n are the entries that represent its position in the n-dimensional space, typically denoted as (x1, x2, ..., xn).
An augmented matrix is a matrix that represents a linear system, including the coefficients of the variables and the constants from the equations.
A basis for a vector space is a set of vectors that are linearly independent and span the entire space.
The Unique Representation Theorem states that for a basis of a vector space, each vector in that space can be expressed uniquely as a linear combination of the basis vectors using a set of scalars.
The operations defined are addition of polynomials and scalar multiplication for any scalar.
Eigenvalues are scalars associated with a linear transformation represented by a matrix, indicating how much the eigenvector is stretched or compressed during the transformation.
A -coordinate vector is a column vector that represents the -coordinates of a vector in relation to a given basis.
1. The system has a unique solution. 2. The system has multiple solutions. 3. The system has no solution (inconsistent system).
The identity transformation is a linear transformation that maps every vector in the vector space to itself, serving as a reference for changing bases.
For vectors in R^n, addition and scalar multiplication work the same way as for vectors in R^3.
The columns of a matrix are the vertical arrays of numbers that represent the components of vectors in a vector space.
The vectors 1, 0, and 0 form the standard basis in a three-dimensional vector space.
A unique solution refers to a single set of values that satisfies all equations in the linear system.
An eigenvector is a non-zero vector that changes by only a scalar factor when a linear transformation is applied to it, corresponding to a specific eigenvalue.
A matrix equation is an equation in which the unknowns are represented as a matrix, and it expresses a relationship between matrices, typically involving multiplication and addition.
Equipping the domain with one basis and the range with another allows for the representation of linear transformations between different coordinate systems in the vector space.
To diagonalize a matrix means to find matrices such that the matrix equation involving a diagonal matrix and an invertible matrix is satisfied.
The final step involves reflecting on the process and outcomes to improve future problem-solving strategies.
The range is the set of all vectors that can be produced as output by the transformation.
They must satisfy two properties: T(u + v) = T(u) + T(v) for all vectors u and v, and T(cu) = cT(u) for all vectors u and all scalars c.
Eigenvalues are scalars associated with a linear transformation that indicate how much the transformation stretches or compresses vectors in the direction of their corresponding eigenvectors.
The characteristic polynomial is the polynomial obtained from the left side of the characteristic equation, which is a polynomial in terms of the eigenvalue.
A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero, making it easier to compute powers and exponentials of matrices.
An equation has non-trivial solutions if there exists at least one solution other than the zero vector.
A simple matrix representation typically refers to a form that is easier to work with, such as a diagonal matrix, which simplifies calculations and understanding of the transformation.
The change-of-coordinates matrix is used to transform the representation of a linear transformation from one basis to another, facilitating the diagonalization process.
The Associative Property states that for any vectors u, v, and w, (u + v) + w = u + (v + w).
Two matrices are said to be similar if one can be transformed into the other by a change of basis, specifically if there exists an invertible matrix such that one matrix is the product of the other and the invertible matrix's inverse.
The axioms of a vector space include closure under addition and scalar multiplication, existence of an additive identity and inverses, and properties like associativity and distributivity.
A linear system is a collection of one or more linear equations involving the same variables, which can be represented in matrix form.
It means that for any vector in the subspace and any scalar, the product of the scalar and the vector is also in the subspace.
Commutativity in vector addition means that for any vectors u and v, u + v = v + u.
Echelon form is a type of matrix form where all non-zero rows are above any rows of all zeros, and the leading coefficient of a non-zero row is to the right of the leading coefficient of the previous row.
The theorem states that any linearly independent set in a vector space with basis must contain no more than the number of vectors in the basis.
The set of all degree polynomials with real coefficients is a subspace of the vector space of all polynomials.
It means that for any two vectors in the subspace, their sum is also in the subspace.
A basis for an eigenspace is a set of linearly independent eigenvectors that span the eigenspace.
A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero, and it represents the linear transformation in a simplified form when the transformation is diagonalizable.
A vector equation is an equation that expresses a vector as a linear combination of other vectors, often used to solve for unknown coefficients.
Multiple solutions indicate that there are infinitely many sets of values that satisfy the equations in the linear system.
A basis is a set of vectors in a vector space that are linearly independent and span the entire space.
The zero vector is the additive identity in a vector space, which, when added to any vector, results in that vector.
The Distributive Property states that for any scalar a and vectors u and v, a(u + v) = au + av.
A vector equation is an equation that expresses a relationship between vectors, often representing a linear combination of vectors equal to another vector.
Scalar multiplication is the operation of multiplying a vector by a scalar (number), resulting in a new vector.
Choosing a basis can simplify the matrix representation of a linear transformation, potentially allowing it to be represented as a diagonal matrix.
If a matrix is not diagonalizable, it means that there are not enough linearly independent eigenvectors to form a basis for the vector space, preventing the matrix from being expressed in diagonal form.
The Elimination Method is a technique used to solve systems of equations by transforming them into equivalent systems through a series of row operations.
An invertible matrix is a matrix that has an inverse, meaning there exists another matrix such that their product is the identity matrix, which is crucial for expressing the linear transformation in terms of a diagonal matrix.
If a linear system is linearly dependent, it implies that at least one of the equations can be expressed as a linear combination of the others, leading to the existence of free variables and non-trivial solutions.
The set of all vectors in the domain that are mapped to the zero vector in the codomain.
Algebraic properties of vectors include operations such as addition, scalar multiplication, and their respective properties like commutativity, associativity, and distributivity.
Scalar multiplication is the operation of multiplying a vector by a scalar, resulting in a vector that is scaled in magnitude but retains its direction.
Reduced echelon form is a matrix form where, in addition to being in echelon form, each leading coefficient is 1 and is the only non-zero entry in its column.
The matrix representation with respect to two bases provides a way to express the linear transformation in terms of the coordinates defined by those bases.
The image of a vector under a transformation is the vector in the range that corresponds to the input vector from the domain.
Row operations are operations performed on the rows of a matrix to simplify it, including row swapping, scaling a row by a non-zero scalar, and adding or subtracting rows.
1. The zero vector is in the subspace. 2. The subspace is closed under addition. 3. The subspace is closed under scalar multiplication.
An inconsistent system is one that has no solution, meaning there are no sets of values that can satisfy all equations simultaneously.
The column vectors of a diagonalizable matrix are the corresponding eigenvectors of the matrix.
The set of all vectors in the codomain that can be expressed as T(v) for some vector v in the domain.
Equivalent systems are systems of equations that have the same solution set, meaning they represent the same geometric object in a given space.
A matrix equation is an equation that involves matrices and vectors, often used to represent systems of linear equations.
The diagonal entries of a diagonalizable matrix are the eigenvalues of the matrix.
If a set of three vectors is linearly dependent, it means at least one of the vectors can be expressed as a linear combination of the others.
It states that for a linear transformation, if the matrix representation with respect to the standard basis is diagonalizable, then it can be expressed as a product of a diagonal matrix and an invertible matrix, with a basis consisting of the column vectors of the invertible matrix.
The set of all lower triangular matrices with real coefficients is a subspace of the vector space of all matrices.
An augmented matrix is a matrix that includes the coefficients of a system of equations along with the constants from the equations, allowing for the application of row operations.
Vectors are linearly independent if no vector in the set can be expressed as a linear combination of the others.
A subspace is a subset of a vector space that is itself a vector space, meaning it is closed under addition and scalar multiplication and contains the zero vector.
An augmented matrix is a matrix that includes the coefficients of the variables and the constants from the equations of a linear system, typically used to solve the system using row operations.
A coordinate vector is a representation of a vector in terms of the basis vectors of a vector space, indicating how much of each basis vector is needed to express the vector.
An eigenspace is one-dimensional if it has a basis consisting of a single eigenvector, meaning that all eigenvectors corresponding to a particular eigenvalue are scalar multiples of that eigenvector.
The set of all matrices with real entries is a vector space.
Finding the coordinate vector involves expressing the vector as a linear combination of the basis vectors and determining the coefficients of this combination.
The standard basis is a set of vectors that are used as a reference for vector spaces, typically consisting of unit vectors along each axis.
