Syllabus: Math 308 – 001 – Winter 2025

identify whether or not a matrix is in Reduced Row Echelon Form,

use Gaussian Elimination to put a matrix into Reduced Row Echelon Form,

use the Reduced Row Echelon Form of an augmented matrix to describe the solution space to a system of linear equations using appropriate notation.

use Pivot Positions to determine whether a linear system is consistent,

determine whether the solution to a consistent linear system is unique.

add vectors,

scale vectors by a real number,

determine whether a vector can be expressed as a linear combination of a set of given vectors.

add matrices,

scale matrices by a real number,

perform matrix-vector multiplication,

perform matrix-matrix multiplication,

translate a system of equations into a matrix equation of the form \(A\mathbf{x} = \mathbf{b}\text{.}\)

determine whether a vector is in the span of a given set of vectors,

describe the span of a set of vectors as a set,

use the span to determine whether a linear system is consistent.

I can determine whether a given set of vectors is linearly independent.

define a matrix transformation,

find the domain and range of a matrix transformation,

compose matrix transformations,

determine whether a given function is a matrix transformation.

use Gaussian Elimination to determine whether a matrix is invertible,

find the inverse of an invertible matrix.

find a basis for a space,

translate between the coordinate systems for different bases.

compute the determinant of a given matrix,

use the determinant to determine whether a matrix is invertible.

find a basis for the null space of a given matrix,

find a basis for the column space of a given matrix,

determine whether a given set satisfies the definition of a vector space.

determine whether a vector is an eigenvector of a given matrix,

find the eigenvalue associated to an eigenvector.

use the characteristic polynomial to find the eigenvalues of a given matrix,

find the algebraic and geometric multiplicities of eigenvalues,

find a basis for the eigenspace associated to an eigenvalue.

determine whether a given matrix is diagonalizable,

diagonalize a diagonalizable matrix.

compute the dot product of two vectors,

use the dot product to find the angle between two vectors,

determine whether two vectors are orthogonal.

compute the transpose of a matrix,

use the transpose to find a basis for the orthogonal complement of a given vector space.

I can perform Orthogonal Decomposition.

use Gram-Schmidt Orthogonalization to find an orthogonal basis for a given vector space,

find an orthonormal basis from an orthogonal basis for a given vector space.