Any page and section numbers in the assignments below refer to the text by Damiano and Little.
Section 1.1: 1, 3, 5, 6, 7, 12.
Section 1.2: 3, 13, 15. And: Prove that in any vector space \(V\), \(0\cdot v = \vec{0}\), for any \(v\in V\).
Section 1.3: 1, 3, 7, 9.
1. Determine if the vector \(v = \begin{pmatrix} 1\\1\\1\\1\end{pmatrix}\) is in the subspace of \(\mathbb{R}^4\) spanned by vectors \(w_1 = \begin{pmatrix} 1\\1\\0\\0\end{pmatrix}\), \(w_2 = \begin{pmatrix} 0\\1\\1\\0\end{pmatrix}\), \(w_3 = \begin{pmatrix} 0\\0\\1\\1\end{pmatrix}\), by setting up and solving a system of linear equations.
2. Now let \(V\) denote the space of all polynomials of degree less than or equal to three with coefficients in \(\mathbb{R}\). Set \(q(x) := 1+x+x^2+x^3\) and \(p_1(x) := 1+x\), \(p_2(x) := x+x^2\), \(p_3(x) := x^2+x^3\). Show that \(q(x)\) is in the subspace of \(V\) spanned by \(p_1(x), p_2(x), p_3(x)\) by setting up a system of linear equations. Solving this system should be easy after working the first problem in this assignment.
Section 1.4: 1(a) – 1(f), 4, 5, 12a.
Section 1.6: 2a, 2b, 2d, 9a, 12, and the following problem: Use Gaussian elimination to prove that any three vectors in \(\mathbb{R}^2\) are linearly dependent.
Section 1.6: 7, 9b, 15.
For this assignment, we will use the trace of a square matrix. We denote this value by \(\mathrm{tr}(A) = a_{11}+a_{22}+\cdots+a_{nn}\). In other words, the trace of a square matrix is the sum of its diagonal entries.
1. Let \(V = \mathrm{M}_2(\mathbb{R})\). Show that the set of matrices \(A\in V\) with \(\mathrm{tr}(A) = 0\) forms a subspace of \(V\). Then find a basis for this subspace.
2. Define \(T: \mathbb{R}^2\to\mathbb{R}^2\) by \(T(a,b) = (3a-2b, -a+b)\). Suppose \(v_1 = (4,5)\), \(v_2 = (-1,3)\) and \(\lambda = 2\). Verify through direct calculation that \(T(v_1+v_2) = T(v_1)+T(v_2)\) and \(T(\lambda v_1) = \lambda T(v_2)\). Here we are using row vectors for ease of notation.
3. Use basic trigonometry to verify that multiplication of column vectors in \(\mathbb{R}^2\) by \(A = \begin{pmatrix}\cos(\theta) & \sin(\theta)\\-\sin(\theta) & \cos(\theta)\end{pmatrix}\) gives rotation counter-clockwise by verifying that \(A\cdot\begin{pmatrix}1\\0\end{pmatrix}\) is the result of rotating \(\begin{pmatrix}1\\0\end{pmatrix}\) by \(\theta\) degrees counter-clockwise and \(A\cdot\begin{pmatrix}0\\1\end{pmatrix}\) is the result of rotating \(\begin{pmatrix}0\\1\end{pmatrix}\) by \(\theta\) degrees counter-clockwise.
Consider the matrices
with \(\lambda\not= 0\). Note that the matrices \(L, M, N\) are obtained from \(A\) by elementary row operations. Let \(C = \begin{pmatrix}\alpha_1\\\alpha_2\\\alpha_3\end{pmatrix}\) be a column vector.
1. Show that \(AC = \vec{0}\) if and only if \(LC = \vec{0}\).
2. Show that \(AC = \vec{0}\) if and only if \(MC = \vec{0}\).
3. Show that \(AC = \vec{0}\) if and only if \(NC = \vec{0}\).
Thus the solution space of \(A\) is the same as the solution space of the row reduced echelon form of \(A\).
Let \(T:\mathbb{R}^3\to\mathbb{R}^2\) and \(S:\mathbb{R}^2\to\mathbb{R}^2\) be the linear transformations
Let \(\alpha\), \(\beta\), \(\gamma\) be the following bases:
1. Compute the matrix of \(T\) with respect to the bases \(\alpha\) and \(\beta\), denoted \([T]_{\beta}^{\alpha}\).
2. Compute the matrix of \(S\) with respect to the bases \(\beta\) and \(\gamma\), denoted \([S]_{\gamma}^{\beta}\).
3. Compute the matrix of \(ST:\mathbb{R}^3\to\mathbb{R}^2\) with respect to the bases \(\alpha\) and \(\gamma\), denoted \([ST]_{\gamma}^{\alpha}\).
4. Verify directly that \([ST]_{\gamma}^{\alpha} = [S]_{\gamma}^{\beta}\,[T]_{\beta}^{\alpha}\).