Course

in the solution to a problem on a quiz, homework, project or exam

you distinguish between a consistent and an inconsistent system of linear equations

you recognize when systems of linear equations are equivalent

you recognize a homogeneous system of equations

you distinguish between the trivial and non-trivial solutions of an homogeneous system of equations

you recognize the connection between solving linear systems and the geometry of intersecting linear objects

in the solution to a problem on a quiz, homework, project or exam

you distinguish between the rows and columns in defining a matrix

you distinguish between row and column vectors

you recognize the dimension of a matrix

you recognize the connection between an incidence matrix and a graph of nodes and edges

in the solution to a problem on a quiz, homework, project or exam

you calculate a scalar multiple of a matrix

you prove the properties of scalar multiplication of a matrix

you calculate the sum of two equal dimension matrices

you prove that matrix addition is commutative

you prove that matrix addition is associative

you recognize when it is possible to multiply two matrices

you calculate the n x k product of an n x m matrix with an m x k matrix

you calculate the product of an n x m matrix with an m x 1 column vector

you interpret the product of an n x m matrix with an m x 1 column vector as a linear combination of the columns of the matrix

you express a linear system of m equations in n variables as a multiplication of an m x n matrix with an m x 1 column vector

you prove that matrix multiplication is associative

you prove that matrix multiplication distributes across matrix addition

you recognize that matrix multiplication is not commutative

you recognize that for matrices AB = AC with A not the zero matrix is not sufficient to deduce that B = C

you recognize that for matrices AB = 0 is not sufficient to deduce that one of the matrices must be the zero matrix

you recognize and form the transpose of a matrix

you prove the properties of matrix transposition

you use technology to perform matrix calculations

in the solution to a problem on a quiz, homework, project or exam

you recognize diagonal, scalar, identity, symmetric and skew symmetric matrices

you recognize partitioned matrices and use block multiplication to multiply partitioned matrices

you distinguish between invertible or nonsingular matrices and singular matrices

you verify when two matrices are inverses

you prove the properties of inverse matrices

you connect inverse matrices to the solution of a linear system of equations

in the solution to a problem on a quiz, homework, project or exam

you recognize that an m x n matrix acts as a linear transformation from n dimensional space to m dimensional space

you recognize the image of a linear transformation of a vector in m dimensional space

you recognize the range of a linear transformation of a vector in m dimensional space

you recognize how to represent rotations, reflections, projections, dilations and contractions as matrix transformations

you recognize the applications of matrix transformations to problems in computer graphics

You will demonstrate your competence:in the solution to a problem on a quiz, homework, project or exam

you recognize when an m x n matrix is in reduced echelon form

you recognize when an m x n matrix is in reduced row echelon form

you recognize the three elementary row operations

you recognize when two equal dimension matrices are equivalent

you recognize a pivot and a pivot column

you prove that every non-zero m x n matrix is equivalent to a row echelon form matrix

you recognize that every non-zero m x n matrix is equivalent to a unique reduced row echelon form matrix

in the solution to a problem on a quiz, homework, project or exam

you solve a linear system of n equations in n variables by using Gaussian elimination on the augmented n x (n+1) matrix to get an equivalent matrix in row echelon form

you solve a linear system of n equations in n variables by using Gauss-Jordan reduction on the augmented n x (n+1) matrix to get an equivalent matrix in reduced row echelon form

you recognize when the row echelon form of the augmented matrix indicates that the linear system of equations is inconsistent

you recognize when the row echelon form of the augmented matrix indicates that the linear system of equations is consistent but has infinitely many solutions

you use the row echelon form of the augmented matrix to construct the form of the solutions when the linear system of equations is consistent but has infinitely many solutions

you prove that an homogeneous system with more variables than equations always has non-trivial solutions

you prove that every solution of a non-homogeneous system is the sum of a particular solution and a solution of the associated homogeneous system

in the solution to a problem on a quiz, homework, project or exam

you recognize an elementary matrix

you recognize that the results of an elementary row operation can be achieved by matrix multiplication (on the left) with the corresponding elementary matrix

you recognize that two matrices are equivalent if and only if one is equal to product of the other with elementary matrices

you recognize that the inverse of every elementary matrix is an elementary matrix of the same type

you recognize that if an nxn matrix A has only the trivial homogeneous solution to Ax = 0, then A is row equivalent to the nxn identity matrix, In

you prove that an nxn matrix A is nonsingular if and only if A is the product of elementary matrices

you prove that an nxn matrix A is nonsingular if and only if A is row equivalent to the nxn identity matrix, In

you prove that for an nxn matrix A, Ax = 0 has nontrivial solutions if and only if A is singular

you prove that for an nxn matrix A, Ax = b has a unique solutions if and only if A is nonsingular

you generate the inverse of a nonsingular nxn matrix by transforming the partitioned matrix [A, In] to reduced row echelon form

you prove that an nxn matrix A is singular if and only if the reduced row echelon form of A contains a row of zeros

you prove that for an nxn matrices A and B, if AB = In, then B is the inverse matrix of A

in the solution to a problem on a quiz, homework, project or exam

you recognize that two matrices are equivalent if and only if one can be obtained from the other by a finite sequence of elementary row or elementary column operations

you prove that for mxn matrices A and B are equivalent if and only if B = PAQ for nonsingular matrices P and Q

in the solution to a problem on a quiz, homework, project or exam

you recognize a permutation as a one to one map from S = {1, 2, ...n} onto S

you recognize an inversion in a permutation

you recognize and compute if a given permutation is odd or even

in the solution to a problem on a quiz, homework, project or exam

you recognize the determinant function of an n x n matrix as a sum over permutations

you use the sum over permutations definition to calculate 2 x 2 and 3 x 3 determinants

in the solution to a problem on a quiz, homework, project or exam

you prove that the determinant of A and A transpose are equal

you prove that if you interchange two rows or columns that the determinant changes sign

you prove that if two rows or columns of A are identical then det(A) = 0

you prove that if A has a row or column of zeros then det(A) = 0

you prove that if you multiply a row or column of A by a scalar k, then the determinant of the resulting matrix is k*det(A)

you prove that if you add any scalar multiple of a row to another row of A the resulting matrix has a determinant = det(A)

you prove that if a matrix is in upper or lower triangular form that the determinant is the product of the diagonal elements

you use the properties of determinants to compute a determinant by using the elementary row or column operations

you prove that if E is an elementary matrix that det(EA) = det(E)det(A) and det(AE) = det(A)det(E)

you prove that A is nonsingular if and only if det(A) is not zero

you prove the homogeneous equation Ax = 0 has non trivial solutions if and only if det(A) = 0

you prove that for square matrices A and B det(AB) = det(A)det(B)

you prove for nonsingular matrices that det(inverse of A) = 1/det(A)

you prove that if A and B are similar matrices that det(A) = det(B)

in the solution to a problem on a quiz, homework, project or exam

you recognize for a square matrix A the minor determinant and cofactor of an element (a)ij

you prove that det(A) = sum over any row (or column) of each element of that row (or column) times the cofactor of that element

you use the cofactor expansion to compute determinants

you apply the computation of determinants to compute areas

in the solution to a problem on a quiz, homework, project or exam

you recognize the adjoint of an n x n matrix A as the matrix whose (i,j) element is the cofactor of (a)ji

you prove that A times the adjoint of A = adjoint of A times A = det(A) In

you prove that if A is nonsingular then the inverse of A = adjoint of A/det(A)

in the solution to a problem on a quiz, homework, project or exam

you prove Cramer's rule for nonsingular matrices

you apply Cramer's rule to the solution of a linear system of equations

you recognize the computational inefficiency of Cramer's rule for a large system of equations

in the solution to a problem on a quiz, homework, project or exam

you add, subtract and scale vectors in two and three dimensions

in the solution to a problem on a quiz, homework, project or exam

you recognize the definition of an abstract vector space with regard to the two operations of vector addition and scalar multiplication

you recognize examples of vector spaces

you recognize the standard symbols for common vector spaces

in the solution to a problem on a quiz, homework, project or exam

you recognize the definition of a subspace of a vector space

you prove that if V is a vector space then the nonempty subset W of V is a subspace if and only if W is closed under both vector addition and scalar multiplication

you recognize subspaces that occur in two and three dimensions

you prove that for an n x n matrix A the solution space of a homogeneous equation Ax = 0 is a subspace of n dimensional space

you recognize the span of a set of vectors in a vector space

you prove that every span in V is a subspace of V

you determine if a particular vector in V is in a particular span

you determine a span for the solutions of a homogeneous equation

in the solution to a problem on a quiz, homework, project or exam

you recognize the definition of linear independence for a set of vectors in a vector space

you determine if a particular set of vectors is linearly independent

you prove that in n dimensional space a set of n vectors is linearly independent if and only if the n x n matrix whose columns are the n vectors is nonsingular

you prove that if S1 is a subset of S2 which is a subset of V, then if S1 is linearly dependent so is S2 and if S2 is linearly independent so is S1

in the solution to a problem on a quiz, homework, project or exam

you recognize that a basis is a linearly independent span

you recognize the natural or standard basis of common vector spaces

you demonstrate whether a given set of vectors form a basis

you distinguish between finite-dimensional and infinite-dimensional vector spaces

you prove that the representation of a vector in a given basis is unique

you prove that every span has a subset which is a basis of that span

you prove that if V is a vector space of dimension n and T is a subset of r linearly independent vectors in V, then r can not be larger than n

you recognize a maximal independent subset of a subset of a vector space

you prove that if S is a linearly independent set of vectors in an n-dimensional vector space V then there is a basis for V that contains S

you prove that any set of n linearly independent vectors in a vector space of dimension n is a basis for that vector space

you prove that any set of n vectors in a vector space of dimension n that spans the vector space is a basis for that vector space

you prove that if S is a span of a vector space V, then a maximal independent subset of S is a basis of V

in the solution to a problem on a quiz, homework, project or exam

you recognize that the null space of a matrix is the subspace of homogeneous solutions

you find a basis for the null space of a matrix

you recognize that for an m x n matrix A, if the reduced row echelon form of the augmented matrix has r nonzero rows, then the dimension of the basis of the null space is n - r

in the solution to a problem on a quiz, homework, project or exam

you recognize an ordered basis and a coordinate vector with respect to an ordered basis

you generate the coordinate vector for a given vector and a given ordered basis

you prove that if two vectors have the same coordinate vector they are equal

you recognize when two vector spaces are isomorphic

you prove that isomorphism is an equivalence relation (i.e, that isomorphism is reflexive, symmetric and transitive)

you prove that every n-dimensional real vector space is isomorphic to n-dimensional Euclidean space

you prove that two finite vector spaces are isomorphic if and only if their dimensions are equal

you recognize the transition matrix for two different ordered bases

you construct the transition matrix for two given ordered bases

you prove that the transition matrix is nonsingular and that the transition matrix from basis S to basis T is the inverse of the transition matrix from T to S

in the solution to a problem on a quiz, homework, project or exam

you recognize the row and column spaces of an m x n vector

you prove for m x n matrices A and B that if A is row (column) equivalent to B, then the row (column) spaces of A and B are equal

you recognize row (column) rank as the dimension of the row (column) space

you construct a basis for subspace of a span

you prove that the row and column rank of a matrix are equal

you prove that two m x n matrices have equal rank if and only if they are equivalent matrices

you prove for an nxm matrix A that the rank of A + dimension of the null space of A = n

you prove that an n x n matrix has rank n if and only if the matrix is row equivalent to the n-dimensional identity matrix

you prove that an n x n matrix has rank n if and only if it is nonsingular

you prove for an n x n matrix A that Ax = 0 has nontrivial solutions if and only if the rank of A < n

you prove for an n x n matrix A that Ax = b has a solution if and only if the rank of A equals the rank of the augmented matrix

you recognize that the following nine statements about an n x n matrix A are logically equivalent : 1. A is nonsingular; 2. the homogeneous equation has only the trivial solution; 3. A is row (column) equivalent to the n-dimensional identity matrix; 4. Ax = b has a unique solution for every n dimensional vector b; 5. A is the product of elementary matrices; 6. det(A) is not zero; 7. The rank of A is n; 8. The dimension of the null space is zero; 9. The rows (columns) of A are linearly independent

in the solution to a problem on a quiz, homework, project or exam

you calculate the length and direction of vectors in two and three dimensional Euclidean space

you calculate the dot product of two vectors in two and three dimensional Euclidean space

you prove the properties of the dot product of two vectors in two and three dimensional Euclidean space

you calculate the cross product of two vectors in three dimensional Euclidean space

you prove the properties of the cross product of two vectors in two and three dimensional Euclidean space

you use vector cross product to calculate areas

you use vector cross product to generate a normal vector to a plane

in the solution to a problem on a quiz, homework, project or exam

you recognize a inner product defined on a vector space

you recognize an inner product as a generalization of the dot product

you recognize that a vector space with an inner product is an inner product space

you recognize that the matrix of inner products of an ordered basis ( the matrix of the inner product with respect to the ordered basis) for a vector space is a symmetric matrix that determines the inner product for every pair of vectors in the vector space

you prove the Cauchy - Schwarz Inequality

you prove the triangle inequality from the Cauchy - Schwarz inequality

you recognize orthogonal and orthonormal vectors

you prove that any set of orthogonal vectors is linearly independent

in the solution to a problem on a quiz, homework, project or exam

you prove that the Gram - Schmidt process generates an orthonormal basis for any finite dimensional subspace of an inner product space

you use the Gram - Schmidt process to construct an orthonormal basis

you prove that the matrix of the inner product with respect to an ordered basis is the identity matrix if the basis is orthonormal

you prove that an m x n matrix with n linearly independent columns has a QR factorization

in the solution to a problem on a quiz, homework, project or exam

you recognize the orthogonal complement to a subspace of an inner product space

you prove that the orthogonal complement of a subspace W of an inner product space V is also a subspace whose intersection with W is the zero vector

you determine a basis for the orthogonal complement of a subspace

you prove for an inner product space V with a finite subspace W that every vector in V is the sum of a vector in W with a vector in the orthogonal complement of W and that this decomposition is unique

you prove for an inner product space V with a finite subspace W that the orthogonal complement of the orthogonal complement of W is W

you prove for an m x n matrix that the null space of A is the orthogonal complement of the row space of A

you prove that if the rank of matrix A is r, then the dimension of the orthogonal complement of the row space of A is n-r

you prove for any m x n matrix A and x any vector in n-dimensional space, that x is given uniquely by x = x1 + x2 where x1 is in the row space of A with b = Ax1 and x2 in the null space of A. In particular Ax = Ax1 = b

you recognize the orthogonal projection of a vector onto a subspace W of an inner product space

you prove that the orthogonal projection of a vector v onto a subspace W of an inner product space is the vector in W closest to v

in the solution to a problem on a quiz, homework, project or exam

you recognize the use of orthogonal projections in Fourier series

you recognize the use of orthogonal projections in generating the least squares solution to Ax = b

in the solution to a problem on a quiz, homework, project or exam

you recognize a linear transformation

you distinguish between a linear operator and a linear transformation

you prove that every matrix transformation is a linear transformation

you prove that the mapping from a vector to its coordinates with respect to an ordered basis is a linear transformation

you prove that a linear transformation on a finite dimensional vector space V is completely determined by its transformation on any basis of V

you construct the standard m x n matrix that represents a linear transformation from n dimensional Euclidean space to m dimensional Euclidean space

in the solution to a problem on a quiz, homework, project or exam

you recognize a one to one linear transformation

you recognize the kernel of a linear transformation L from V to W as the subspace of V that L maps to the zero vector of W

you prove that a linear transformation is one to one if and only if the kernel of the transformation consists only of the zero vector of V

you determine the kernel of a given linear transformation

you recognize the range (or image) of a linear transformation L from V to W as the subspace of W whose elements are the result of applying L to an element of V

you recognize that a linear transformation L from V to W is onto if the range of L = W

you determine if a given linear transformation is one to one

you determine if a given linear transformation is onto

you prove that for a linear transformation L from an n -dimensional vector space V to W that the dimension of the kernel of L + the dimension of the range of L = n

you prove for a linear transformation from an n-dimensional vector space V to an n-dimensional vector space W that the linear transformation is onto if and only if it is one to one

you prove for a linear transformation from a vector space V to a vector space W that the linear transformation is one to one if and only if the image of every set of linearly independent vectors in V is a linearly independent set of vectors in W

you prove for a linear transformation from an n-dimensional vector space V to an n-dimensional vector space W that the linear transformation is onto and thus one to one if and only if the image of a basis in V is a basis in W

in the solution to a problem on a quiz, homework, project or exam

you recognize for a linear transformation from an n-dimensional vector space V to an m-dimensional vector space W that from ordered bases in both vector spaces there exists a unique m x n matrix which performs the linear transformation via matrix multiplication

you construct the matrix that represents a linear transformation with respect to two given ordered bases

you prove for linear transformation from an n-dimensional vector space V to an n-dimensional vector space W that the linear transformation is onto and one to one if and only if the matrix which represents the transformation with respect to a pair of ordered bases is nonsingular

in the solution to a problem on a quiz, homework, project or exam

you recognize what is meant by the sum of two linear transformations

you recognize what is meant by scalar multiplication of a linear transformation

you prove that the set of all linear transformations from an n-dimensional vector space V to an m-dimensional vector space W is a vector space

you prove that the vector space of all linear transformations from an n-dimensional vector space V to an m-dimensional vector space W is an isomorphism of the vector space of m x n matrices

you prove if L1 is a linear transformation from an m-dimensional vector space V1 to an n-dimensional vector space V2 and L2 is a linear transformation from V2 to a p-dimensional vector space V3 then with respect to chosen ordered bases in each vector space the m x p matrix which represents the composition of L1 with L2 is the matrix product of the m x n matrix which represents L1 with the n x p matrix that represents L2

in the solution to a problem on a quiz, homework, project or exam

you compute the new matrix representation of a linear transformation from a vector space V to a vector space W with respect to a change of given ordered bases in V and W by matrix multiplication of the old representation matrix with transition matrices, in particular: (Q inverse) ( A ) (P) where Q is the transition matrix for a change of basis in W and P is the transition matrix for a change of basis in V

you recognize that for a linear operator on vector space that a change of ordered basis in the matrix representation of the operator is given by the similarity transformation (P inverse) ( A ) (P)

you recognize that the dimension of the range of a linear transformation is the rank of a representation matrix of that linear transformation

you recognize for a linear transformation from a vector space V to a vector space W that the rank of a representation matrix + dimension of the null space of a representation matrix = dimension of V

you recognize that all representation matrices of a given linear operator are similar

you prove that two matrices are similar if and only if they both represent the same linear operator with respect to two bases

you prove that similar matrices have equal rank

in the solution to a problem on a quiz, homework, project or exam

you recognize that the use of homogeneous coordinates enables the use of matrix multiplication to manipulate images in computer graphics applications

you recognize that homogeneous coordinates enable the representation of scalings, projections, rotations and translations in two and three dimensions

in the solution to a problem on a quiz, homework, project or exam

you recognize that an eigenvector of a linear operator L on a vector space V is a nonzero vector x in V with the property that L(x) = scalar constant (called an eigenvalue) times x

you recognize that eigenvalues can be complex numbers and the eigenvector can have complex components

you construct the characteristic polynomial that is associated with an eigenvalue problem

you determine the eigenvalues by determining the roots of the characteristic polynomial

you determine an eigenvector by finding the nontrivial solutions of the associated homogeneous equation

you find the eigenvalues and associated eigenvectors for 2 x 2 and 3 x 3 matrices

you recognize that equivalent matrices do not necessarily have the same eigenvalues

in the solution to a problem on a quiz, homework, project or exam

you recognize that a linear operator on a finite vector space is diagonalizable if there exits an ordered basis on V for which the representation matrix of L is a diagonal matrix

you prove that similar matrices have the same eigenvalues

you prove that a linear operator on a finite vector space is diagonalizable if and only if V has a basis of eigenvectors of L

you prove that if a linear operator on a finite vector space has eigenvectors which form a basis of V then the representation of L with respect to this basis is a diagonal matrix with the eigenvalues of L as the diagonal elements of the matrix

you prove that an n x n matrix is similar to a diagonal matrix if and only if the matrix has n linearly independent eigenvectors

you construct the similarity transformation that would diagonalize an n x n matrix with n linearly independent eigenvectors

you prove that if the n roots of the characteristic equation associated with an n x n matrix are distinct that the matrix can be diagonalized and the n eigenvectors are linearly independent

you analyze the eigenvectors of an n x n matrix if there is a degeneracy in the eigenvalue spectrum, i.e., if there are multiple roots of the characteristic equation

in the solution to a problem on a quiz, homework, project or exam

you prove for a symmetric matrix that all of the eigenvalues are real

you prove that for a symmetric matrix eigenvectors associated with different eigenvalues are orthogonal

you recognize that an orthogonal matrix has the property that its inverse is its transpose

you prove that a matrix is orthogonal if and only if its columns and rows are orthonormal

you prove that any symmetric matrix can be diagonalized by a similarity transformation using orthogonal matrices

you recognize that there a k linearly independent eigenvectors corresponding to root of multiplicity k of the characteristic polynomial associated with a symmetric matrix

you recognize that if it is necessary to deal with a degeneracy in the eigenvalues a Gram - Schmidt process can be used to ensure that an orthonormal set of n eigenvectors can be constructed for any n x n symmetric matrix

you diagonalize 2 x 2 and 3 x 3 symmetric matrices

you recognize the differences between diagonalizing a nonsymmetric square matrix and a square matrix

in the solution to a problem on a quiz, homework, project or exam

you find the singular values of a matrix

you determine the singular value decomposition of a matrix

you determine the dominant eigenvalue of a matrix

you determine a bound on the absolute value of the dominant eigenvalue of a matrix

you apply eigenvalues to the solution of linear differential equations

you apply eigenvalues to characterize equilibrium points of dynamical systems