We define the determinant of a square matrix in terms of cofactor expansion along the first row.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0010/main

We define the determinant of a square matrix in terms of cofactor expansion along the first column, and show that this definition is equivalent to the definition in terms of cofactor expansion along the first row.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0020/main

We examine the effect of elementary row operations on the determinant and use row reduction algorithm to compute the determinant.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0030/main

We summarize the properties of the determinant that we already proved, and prove that a matrix is singular if and only if its determinant is zero, the determinant of a product is the product of the determinants, and the determinant of the transpose is equal to the determinant of the matrix.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0040/main

We derive the formula for Cramer’s rule and use it to express the inverse of a matrix in terms of determinants.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0060/main

We interpret a 2×2 determinant as the area of a parallelogram, and a 3×3 determinant as the volume of a parallelepiped.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0070/main

We explore the theory behind finding the eigenvalues and associated eigenvectors of a square matrix.https://ximera.osu.edu/la/LinearAlgebra/EIG-M-0020/main

In this module we discuss algebraic multiplicity, geometric multiplicity, and their relationship to diagonalizability.https://ximera.osu.edu/la/LinearAlgebra/EIG-M-0050/main

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 prove several results concerning linear independence of rows and columns of a matrix.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0110/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