## MTH 236 Section Objectives

##### Chapter One, Section I
1. Use elementary row operations (Gauss's method) to find the solution set of a linear system of equations.
2. Identify the echelon form of a system. Identify leading and free variables.
3. Write a linear system in matrix form and apply Gauss's method in matrix form.
4. Use vectors to describe the solution set of a linear system (General = Particular + Homogeneous).
5. Determine whether a linear system has no solution, exactly one solution, or infinitely many solutions.
6. Determine whether a square matrix is singular or nonsingular.
##### Chapter One, Section III
1. Identify the reduced row echelon form (RREF) of a matrix
2. Use Gauss-Jordan elimination to find the RREF of a matrix.
3. Determine if two matrices are row equivalent.
4. Use the RREF of an augmented matrix to find the solution set of the corresponding linear system.
5. Determine if a relation is an equivalence relation.
##### Chapter Two, Section I
1. Determine whether a set with operations is a vector space.
2. Prove properties in vector spaces.
3. Perform operations in vector spaces.
4. Determine if a subset of a vector space is a subspace.
5. Determine if a vector is in the span of a subspace.
6. Find vectors that span a subspace and parameterize the subspace's description.
##### Chapter Two, Section II
1. Show that vectors are linearly dependent/independent.
2. Explain what can happen to the span of a set when vectors are added to or removed from the set.
##### Chapter Two, Section III
1. Find a basis for a vector space.
2. Show that is subset is a basis.
3. Given a basis for a vector space, find the basis representation for a given vector.
4. Find the dimension of a vector space.
5. Reduce a linearly dependent spanning set to a basis.
6. Determine a basis for the row space of a matrix. Determine row rank of a matrix.
7. Determine a basis for the column space of a matrix. Determine column rank of a matrix.
8. Determine the transpose of a matrix.
9. Know the relationship between row rank and column rank.
10. Know and apply the various ways to think about a nonsingular matrix.
##### Chapter Three, Section I
1. Determine is a function is one-to-one.
2. Determine if a function is onto.
3. Determine if a map between vector spaces is an isomorphism.
4. Determine if vector spaces are isomorphic.
5. Know, explain, and apply that any $n$$n$-dimensional vector space is isomorphic to ${\mathbb{R}}^{n}$$\mathbb{R}^n$.
##### Chapter Three, Section II
1. Determine if a map between vector spaces is a homomorphism.
2. Find the unique homomorphism $h:V\to W$$h:V \to W$ that maps the vectors of basis $B=⟨{b}_{1},{b}_{2},\dots ,{b}_{n}⟩$$B=\langle b_1, b_2, \dots, b_n \rangle$ to the vectors $\left\{{w}_{1},{w}_{2},\dots ,{w}_{n}\right\}$$\{w_1, w_2, \dots, w_n\}$, respectively.
3. Know, explain, and apply that a homomorphism is uniquely determined by its action on a basis. (See #2 above).
4. Find the range space and rank of a homomorphism.
5. Find the null space and nullity of a homomorphism.
6. Know, explain, and apply that the image of a subspace under a homomorphism is a subspace.
##### Chapter Three, Section III
1. Find and apply the matrix representation of a homomorphism.
##### Chapter Three, Section IV
1. Perform operations (addition, scalar multiplication, and matrix multiplication) on matrices.
2. Understand and apply the properties (associative, commutative, distributive) of matrix operations when appropriate.
3. Explain matrix multiplication by using various models (dot product, linear combo of columns, linear combo of rows, etc.)
4. Explain that matrix multiplication is essentially a composition of homomorphisms.
5. Know the difference between a left and right inverse.
6. Find the inverse of a matrix and use the properties of inverse matrices.
7. Know several ways to determine when a matrix is invertible.
##### Chapter Three, Section V
1. Find a change-of-basis matrix.
2. Know the properties of the change-of-basis matrix.
##### Chapter Three, Section VI
1. Compute vector projections.
2. Use the Gram-Schmidt process to orthogonalize a basis.
3. Show that a set of nonzero orthogonal vectors is linearly independent.
##### Chapter Four, Section I
1. Know and use the properties of determinants.
2. Use properties to prove properties.
3. Use row operations to compute a determinant.
##### Chapter Four, Section III
1. Use the Laplace expansion to compute a determinant.
2. Know and use properties of the Laplace expansion.
3. Compute the adjoint of a matrix.
4. Use the matrix adjoint to compute an inverse matrix.
5. Use Cramer's rule to solve a system.
##### Chapter Five, Section II
1. Know what it means for matrices to be similar.
2. Given a homomorphism $h:V\to V$$h:V \to V$, and bases $B$$B$ and $D$$D$ for $V$$V$, construct the matrix representations ${\text{Rep}}_{D,D}\left(h\right)$$\mbox{Rep}_{D,D}(h)$, ${\text{Rep}}_{B,B}\left(h\right)$$\mbox{Rep}_{B,B}(h)$, ${\text{Rep}}_{D,B}\left(id\right)$$\mbox{Rep}_{D,B}(id)$, and ${\text{Rep}}_{B,D}\left(id\right)$$\mbox{Rep}_{B,D}(id)$, and thereby illustrate the similarity.
3. Know and demonstrate what it means for a matrix to be diagonalizable.
4. Determine the characteristic polynomial of a matrix.
5. Find the eigenvalues and eigenvectors of a matrix.
6. For each eigenvalue, determine the corresponding eigenspace of a matrix.
7. Diagonalize a matrix.
8. Compute the matrix exponential of a diagonalizable matrix.
9. Use induction to prove that eigenvectors associated with distinct eigenvalues are linearly independent.
##### Inner Products
1. Determine whether a given "product" defined on a vector space is an inner product.
2. Show that vectors are orthogonal with respect to an inner product.
3. Find the norm of a vector in an inner product space.
##### Other general objectives
1. Use induction in proofs.

Last updated April 23, 2024

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