We define a linear transformation from R^n into R^m and determine whether a given transformation is linear.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0010/main

We establish that every linear transformation of R^n is a matrix transformation, and define the standard matrix of a linear transformation.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0020/main

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 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 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

We define a homogeneous linear system and express a solution to a system of equations as a sum of a particular solution and the general solution to the associated homogeneous system.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0050/main

We define closure under addition and scalar multiplication, and we demonstrate how to determine whether a subset of vectors in R^n is a subspace of R^n.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0020/main

We define bases and consider examples of bases of Rn and subspaces of Rn.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0030/main

We discuss existence of bases of R^n and subspaces of R^n, and define dimension.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0035/main

We define the row space, the column space, and the null space of a matrix, and we prove the Rank-Nullity Theorem.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0040/main

We state the definition of an abstract vector space, and learn how to determine if a given set with two operations is a vector space. We define a subspace of a vector space and state the subspace test. We find linear combinations and span of elements of a vector space.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0050/main

We revisit the definitions of linear independence, bases, and dimension in the context of abstract vector spaces.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0060/main