- Use elementary row operations (Gauss's method) to find the solution set of a linear system of equations.
- Identify the echelon form of a system. Identify leading and free variables.
- Write a linear system in matrix form and apply Gauss's method in matrix form.
- Use vectors to describe the solution set of a linear system (General = Particular + Homogeneous).
- Determine whether a linear system has no solution, exactly one solution, or infinitely many solutions.
- Determine whether a square matrix is singular or nonsingular.

- Identify the reduced row echelon form (RREF) of a matrix
- Use Gauss-Jordan elimination to find the RREF of a matrix.
- Determine if two matrices are row equivalent.
- Use the RREF of an augmented matrix to find the solution set of the corresponding linear system.
- Determine if a relation is an equivalence relation.

- Determine whether a set with operations is a vector space.
- Prove properties in vector spaces.
- Perform operations in vector spaces.
- Determine if a subset of a vector space is a subspace.
- Determine if a vector is in the span of a subspace.
- Find vectors that span a subspace and parameterize the subspace's description.

- Show that vectors are linearly dependent/independent.
- Explain what can happen to the span of a set when vectors are added to or removed from the set.

- Find a basis for a vector space.
- Show that is subset is a basis.
- Given a basis for a vector space, find the basis representation for a given vector.
- Find the dimension of a vector space.
- Reduce a linearly dependent spanning set to a basis.
- Determine a basis for the row space of a matrix. Determine row rank of a matrix.
- Determine a basis for the column space of a matrix. Determine column rank of a matrix.
- Determine the transpose of a matrix.
- Know the relationship between row rank and column rank.
- Know and apply the various ways to think about a nonsingular matrix.

- Determine is a function is one-to-one.
- Determine if a function is onto.
- Determine if a map between vector spaces is an isomorphism.
- Determine if vector spaces are isomorphic.
- Know, explain, and apply that any
-dimensional vector space is isomorphic to$n$ .${\mathbb{R}}^{n}$

- Determine if a map between vector spaces is a homomorphism.
- Find the unique homomorphism
that maps the vectors of basis$h:V\to W$ to the vectors$B=\u27e8{b}_{1},{b}_{2},\dots ,{b}_{n}\u27e9$ , respectively.$\{{w}_{1},{w}_{2},\dots ,{w}_{n}\}$ - Know, explain, and apply that a homomorphism is uniquely determined by its action on a basis. (See #2 above).
- Find the range space and rank of a homomorphism.
- Find the null space and nullity of a homomorphism.
- Know, explain, and apply that the image of a subspace under a homomorphism is a subspace.

- Find and apply the matrix representation of a homomorphism.

- Perform operations (addition, scalar multiplication, and matrix multiplication) on matrices.
- Understand and apply the properties (associative, commutative, distributive) of matrix operations when appropriate.
- Explain matrix multiplication by using various models (dot product, linear combo of columns, linear combo of rows, etc.)
- Explain that matrix multiplication is essentially a composition of homomorphisms.
- Know the difference between a left and right inverse.
- Find the inverse of a matrix and use the properties of inverse matrices.
- Know several ways to determine when a matrix is invertible.

- Find a change-of-basis matrix.
- Know the properties of the change-of-basis matrix.

- Compute vector projections.
- Use the Gram-Schmidt process to orthogonalize a basis.
- Show that a set of nonzero orthogonal vectors is linearly independent.

- Know and use the properties of determinants.
- Use properties to prove properties.
- Use row operations to compute a determinant.

- Use the Laplace expansion to compute a determinant.
- Know and use properties of the Laplace expansion.
- Compute the adjoint of a matrix.
- Use the matrix adjoint to compute an inverse matrix.
- Use Cramer's rule to solve a system.

- Know what it means for matrices to be similar.
- Given a homomorphism
, and bases$h:V\to V$ and$B$ for$D$ , construct the matrix representations$V$ ,${{\textstyle \text{Rep}}}_{D,D}(h)$ ,${{\textstyle \text{Rep}}}_{B,B}(h)$ , and${{\textstyle \text{Rep}}}_{D,B}(id)$ , and thereby illustrate the similarity.${{\textstyle \text{Rep}}}_{B,D}(id)$ - Know and demonstrate what it means for a matrix to be diagonalizable.
- Determine the characteristic polynomial of a matrix.
- Find the eigenvalues and eigenvectors of a matrix.
- For each eigenvalue, determine the corresponding eigenspace of a matrix.
- Diagonalize a matrix.
- Compute the matrix exponential of a diagonalizable matrix.
- Use induction to prove that eigenvectors associated with distinct eigenvalues are linearly independent.

- Determine whether a given "product" defined on a vector space is an inner product.
- Show that vectors are orthogonal with respect to an inner product.
- Find the norm of a vector in an inner product space.

- Use induction in proofs.

*Last updated April 23, 2024*