We introduce standard unit vectors in R^2, R^3 and R^n, and express a given vector as a linear combination of standard unit vectors.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0035/main
We define the dot product and prove its algebraic properties.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0050/main
We state and prove the cosine formula for the dot product of two vectors, and show that two vectors are orthogonal if and only if their dot product is zero.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0060/main
We find the projection of a vector onto a given non-zero vector, and find the distance between a point and a line.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0070/main
We define the cross product and prove several algebraic and geometric properties.https://ximera.osu.edu/la/LinearAlgebra/VEC-M-0080/main
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 state and prove the Laplace Expansion Theorem for determinants.https://ximera.osu.edu/la/LinearAlgebra/DET-M-0050/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 introduce the concepts of eigenvalues and eigenvectors of a matrix.https://ximera.osu.edu/la/LinearAlgebra/EIG-M-0010/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