Resources
Examinations
Midterm 1
Midterm 2
Midterm 3
Final Examination
Assignments
Assignment 1 - Due January 18
Do the following problems from the text.
| Section | Problems |
|---|---|
| 1.1 | 11-14, 20, 39-42 |
| 1.2 | 1, 7, 11, 21, 45 |
| 1.3 | 5, 13, 18, 34 |
| 1.4 | 12, 21-22 |
Assignment 2 - Due January 25
Do the following problems from the text.
| Section | Problems |
|---|---|
| 1.5 | 5, 6, 11, 12, 23 |
| 1.7 | 1, 2, 7, 8, 11 |
The following problems require formal proofs.
Problem A: Let \(X\) and \(Y\) be finite subsets of \(\mathbb{R}^n\). Prove that \(\operatorname{Span}(X \cap Y) \subseteq \operatorname{Span}(X) \cap \operatorname{Span}(Y)\).
Problem B: Let \(\mathbf{u},\mathbf{v},\mathbf{w}\) be column vectors in \(\mathbb{R}^n\). Prove that \(\{\mathbf{u},\mathbf{v},\mathbf{w}\}\) is linearly independent if and only if \(\{\mathbf{u}+\mathbf{v},\mathbf{v}+\mathbf{w},\mathbf{u}+\mathbf{w}\}\) is linearly independent.
Assignment 3 - Due February 13
Do the following problems from the text.
| Section | Problems |
|---|---|
| 1.8 | 4, 17 |
| 1.9 | 16, 22, 37, 38 |
| 2.1 | 2, 9, 10, 12 |
The following problems require formal proofs.
Problem A: Let \(m\) and \(b\) be real numbers. Let \(f : \mathbb{R}^1 \to \mathbb{R}^1\) be the function given by \(f(x)=mx+b\). Prove that \(f\) is a linear transformation if and only if \(b=0\).
Problem B: Let \(A\) be a fixed \(n \times n\) matrix. Prove that \(AB=BA\) for all \(n \times n\) matrices \(B\) if and only if \(A = \lambda I_n\) for some real number \(\lambda\).
Assignment 4 - Due February 15
Do the following problems from the text.
| Section | Problems |
|---|---|
| 2.2 | 9, 39, 40, 41, 42 |
| 2.3 | 1, 2, 3, 4, 5, 6 |
The following problems require formal proofs.
Problem A: Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix. Prove that \((AB)^T=B^TA^T\).
Problem B: Prove that the matrix \(A = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\) is invertible if and only if \(ad-bc \ne 0\).
Assignment 5 - Due February 22
Do the following problems from the text.
| Section | Problems |
|---|---|
| 3.1 | 2, 4, 10, 14 |
| 3.2 | 6, 8, 12, 14, 46 |
The following problems require formal proofs.
Problem A: Let \(A\) be an invertible matrix. Prove that \(\det(A^{-1})=\det(A)^{-1}\).
Problem B: Let \(A\) be an \(n \times n\) matrix and let \(r\) be a scalar. Prove that \(\det(rA)=r^n\det(A)\).
Assignment 6 - Due March 14
Do the following problems from the text.
| Section | Problems |
|---|---|
| 3.3 | 12, 22, 24 |
| 4.1 | 10, 12, 16, 18 |
The following problems require formal proofs.
Problem A: A matrix \(A\) is symmetric if \(A=A^T\). Prove that the set of symmetric matrices form a subspace of the vector space of all \(n \times n\) matrices.
Problem B: Let \(U,V\) be subspaces of a vector space \(W\). Prove that \(U \cap V\) is a subspace of \(W\). Find an example where \(U \cup V\) is not a subspace of \(W\).
Assignment 7 - Due March 21
Do the following problems from the text.
| Section | Problems |
|---|---|
| 4.2 | 2, 6, 24 |
| 4.3 | 2, 4, 8, 14, 16 |
The following problems require formal proofs.
Problem A: Let \(T : V \to W\) be a linear transformation from a vector space \(V\) to a vector space \(W\). Prove that the range of \(T\) is a subspace of \(W\).
Problem B: Let \(T : V \to W\) be a linear transformation from a vector space \(V\) to a vector space \(W\). Suppose \(v_1,\ldots, v_p\) is a linearly independent set in \(V\). Prove that, if \(T\) is injective, then \(T(v_1),\ldots, T(v_p)\) is linearly independent.
Problem C: Let \(T : V \to W\) be a linear transformation from a vector space \(V\) to a vector space \(W\). Suppose \(v_1,\ldots, v_p\) spans \(V\). Prove that, if \(T\) is surjective, then \(T(v_1),\ldots, T(v_p)\) spans \(W\).
Problem D (optional): Let \(a_1,\ldots,a_n\) be distinct real numbers. For each \(1 \le i \le n\), prove that there is a unique polynomial \(f_i\) of degree \(n-1\) such that \(f_i(a_j)=\delta_{ij}\) for all \(1 \le j \le n\). (Here \(\delta_{ij}\) is the Kronecker delta.)
Assignment 8 - Due March 28
Do the following problems from the text.
| Section | Problems |
|---|---|
| 4.3 | 36, 44 |
| 4.4 | 4, 8, 14, 32, 36 |
| 4.5 | 2, 4, 8 |
The following problems require formal proofs.
Problem A: Let \(T : V \to W\) be an invertible linear transformation from a vector space \(V\) to a vector space \(W\). Prove that if \(v_1,\ldots, v_p\) is a basis for \(V\), then \(T(v_1),\ldots, T(v_p)\) is a basis for \(W\).
Problem B: Let \(\mathcal{B}\) be a finite basis for a vector space \(V\). Prove that the vectors \(u_1,\ldots, u_p\) in \(V\) are linearly independent if and only if the coordinate vectors \([u_1]_{\mathcal{B}}, \ldots, [u_p]_{\mathcal{B}}\) are linearly independent.
Problem C: Let \(U,V,W\) be vector spaces and let \(T: U \to V\) and \(S: V \to W\) be linear transformations. Prove that if \(\operatorname{im}(T) \cap \operatorname{ker}(S) = \{0\}\) then \(\operatorname{ker}(T) = \operatorname{ker}(S \circ T)\).
Problem D (optional): Let \(a_1,\ldots, a_n\) be distinct real numbers. Recall from Assignment 7 that, for each \(1 \le i \le n\), there is a unique polynomial \(f_i\) of degree \(n-1\) such that \(f_i(a_j)=\delta_{ij}\) for all \(1 \le j \le n\). Prove that \(\mathcal{B} = \{ f_1,\ldots, f_n \}\) is a basis for the vector space \(\mathbb{P}_{n-1}\) of polynomials of degree \(\le n-1\).
Assignment 9 - Due April 11
Do the following problems from the text.
| Section | Problems |
|---|---|
| 4.6 | 2, 6, 8, 16 |
| 5.4 | 2, 6, 8, |
| 5.1 | 2, 4, 10, 14 |
The following problems require formal proofs.
Problem A: Let \(A\) and \(B\) be \(n \times n\) matrices. Prove that if \(A\) is similar to \(B\) and \(A\) is invertible, then \(B\) is invertible and \(A^{-1}\) is similar to \(B^{-1}\).
Problem B: Let \(M_n\) be the set of \(n \times n\) matrices. Define a relation \(\sim\) on \(M_n\) where \(A \sim B\) means \(A\) is similar to \(B\). Prove that \(\sim\) is an equivalence relation.
Assignment 10 - Due April 18
Do the following problems from the text.
| Section | Problems |
|---|---|
| 5.2 | 4, 8, 12, 16 |
| 5.3 | 2, 6, 10, 14, 20 |
The following problems require formal proofs.
Problem A: Let \(A\) be an \(n \times n\) matrix. Prove that if \(\lambda\) is an eigenvalue of \(A\), then \(\lambda\) is an eigenvalue of \(A^T\).
Problem B: Let \(A\) be an invertible \(n \times n\) matrix. Prove that if \(A\) is diagonalizable, then \(A^{-1}\) is diagonalizable.