We establish that a plane is determined by a point and a normal vector, and use this information to derive a general equation for planes in R^3.https://ximera.osu.edu/la/LinearAlgebra/RRN-M-0030/main

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 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 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 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 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