We define linear transformation for abstract vector spaces, and illustrate the definition with examples.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0022/main

We establish that a linear transformation of a vector space is completely determined by its action on a basis.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0025/main

We define composition of linear transformations, inverse of a linear transformation, and discuss existence and uniqueness of inverses.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0030/main

We prove that a linear transformation has an inverse if and only if the transformation is “one-to-one” and “onto”.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0035/main

We define the image and kernel of a linear transformation and prove the Rank-Nullity Theorem for linear transformations.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0050/main

We define isomorphic vector spaces, discuss isomorphisms and their properties, and prove that any vector space of dimension n is isomorphic to R^n.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0060/main

We find standard matrices for classic transformations of the plane such as scalings, shears, rotations and reflections.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0070/main

We find the matrix of a linear transformation with respect to arbitrary bases, and find the matrix of an inverse linear transformation.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0080/main

We introduce matrices, define matrix addition and scalar multiplication, and prove properties of those operations.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0010/main

We introduce matrix-vector and matrix-matrix multiplication, and interpret matrix-vector multiplication as linear combination of the columns of the matrix.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0020/main

We define the transpose of a matrix and state several properties of the transpose. We introduce symmetric, skew symmetric and diagonal matrices.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0025/main

We interpret linear systems as matrix equations and as equations involving linear combinations of vectors. We define singular and nonsingular matrices.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0030/main

We develop a method for finding the inverse of a square matrix, discuss when the inverse does not exist, and use matrix inverses to solve matrix equations.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0050/main

We introduce elementary matrices and demonstrate how multiplication of a matrix by an elementary matrix is equivalent to to performing an elementary row operation on the matrix.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0060/main

We solve systems of equations in two and three variables and interpret the results geometrically.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0010/main

We introduce the augmented matrix notation and solve linear system by carrying augmented matrices to row-echelon or reduced row-echelon form.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0020/main

We introduce Gaussian elimination and Gauss-Jordan elimination algorithms, and define the rank of a matrix.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0030/main