We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0040/main

We define the span of a collection of vectors and explore the concept algebraically and geometrically.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0090/main

We define linear independence of a set of vectors, and explore this concept algebraically and geometrically.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0100/main

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