Math 540 - Additional Homework Problems

Acknowledgement. Some of these problems (sometimes in slightly modified form) are from Linear Algebra Done Wrong, S. Treil, © 2004, 2009.

Problem # X.1: (a) Check that the vectors \(\mathbf{v}=(1,2,1)\) and \(\mathbf{w}=(8,4,-1)\) are solutions to the linear equation \[ 2x_1 - 3x_2 + 4x_3 = 0. \qquad\text{(1)} \]
(b) The vector \(\mathbf{u}=(-37,-14,8)\) is also a solution to equation (1). Find real numbers \(a,b\in\mathbf{R}\) so that \[ \mathbf{u} = a\mathbf{v}+b\mathbf{w}. \]
(c) Prove the following general result. If the vector \(\mathbf{z}=(z_1,z_2,z_3)\in\mathbf{R}^3\) is a solution to equation (1), then there are scalars \(a,b\in\mathbf{R}\) so that \[ \mathbf{z} = a\mathbf{v}+b\mathbf{w}. \qquad\text{(2)} \]
(d) In (c), prove that for a given vector \(\mathbf{z}\), there is only one choice for \(a\) and \(b\) that makes equation (2) true.

Problem # X.2: An \(m\)-by-\(n\) matrix with coefficients in a field \(\mathbb{F}\) is defined to be an \(m\)-by-\(n\) array of elements of \(\mathbb{F}\). We write \(M_{m,n}(\mathbb{F})\) for the set of all such matrices, so an element \(A\in M_{m,n}(\mathbb{F})\) looks like \[ A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{pmatrix} \] We make \(M_{m,n}(\mathbb{F})\) into a vector space in the obvious way: \[ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \\ \end{pmatrix} + \begin{pmatrix} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{m1} & \cdots & b_{mn} \\ \end{pmatrix} = \begin{pmatrix} a_{11}+b_{11} & \cdots & a_{1n}+b_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1}+b_{m1} & \cdots & a_{mn}+b_{mn} \\ \end{pmatrix} \] and \[ c \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \\ \end{pmatrix} = \begin{pmatrix} ca_{11} & \cdots & ca_{1n} \\ \vdots & \ddots & \vdots \\ ca_{m1} & \cdots & ca_{mn} \\ \end{pmatrix} \]
(a) Write down a basis for the \(\mathbb{F}\)-vector space \(M_{3,2}(\mathbb{F})\) of 3-by-2 matrices. What is the dimension of \(M_{3,2}(\mathbb{F})\)?
(b) More generally, what is the dimension of \(M_{m,n}(\mathbb{F})\)?

Problem # X.3: The transpose of a (square) matrix \(A\), denoted \(A^*\), is obtained by flipping the entries across the main diagonal. So for example \[ \begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix}^* = \begin{pmatrix} 1&4&7\\ 2&5&8\\ 3&6&9\\ \end{pmatrix}. \] A matrix \(A\) is symmetric if \(A^*=A\) and it is anti-symmetric if \(A^*=-A\).
(a) Prove that the set of \(n\)-by-\(n\) symmetric matrices is a vector subspace of \(M_{n,n}(\mathbb{F})\).
(b) Find a basis for the space of 2-by-2 symmetric matrices. What is its dimension?
(c) Generalize by describing a basis for the space of \(n\)-by-\(n\) symmetric matrices and computing its dimension. (It might help to start with 3-by-3.)
(d) Prove that the set of \(n\)-by-\(n\) anti-symmetric matrices is also a vector subspace of \(M_{n,n}(\mathbb{F})\), describe a basis, and compute its dimension.
(e) (Bonus) Let's write \(M_{n,n}(\mathbb{F})^{\text{sym}}\) for the space of symmetric matrices and \(M_{n,n}(\mathbb{F})^{\text{anti-sym}}\) for the space of anti-symmetric matrices. Prove that \[ M_{n,n}(\mathbb{F}) = M_{n,n}(\mathbb{F})^{\text{sym}} + M_{n,n}(\mathbb{F})^{\text{anti-sym}}. \] Is this a direct sum of vector spaces?

Problem # X.4: Find the matrix associated to each of the following linear transformations relative to the given bases.
(a) \(T:\mathbb{F}^3\to\mathbb{F}^3\) defined by \(T(x,y,z)=(2x-3y,3x+5z,7x+2y-z)\), using the standard basis for \(\mathbb{F}^3\).
(b) \(T:\mathbb{F}^2\to\mathbb{F}^3\) defined by \(T(x,y,z)=(2x-3y,3x+5y,-x)\), using the standard bases for \(\mathbb{F}^2\) and \(\mathbb{F}^3\).
(c) Let \(\mathcal{P}_n(\mathbb{F})\) be the \(\mathbb{F}\)-vector space of polynomials of degree at most \(n\) with coefficients in \(\mathbb{F}\). Use the basis \(\{1,t,t^2,\ldots,t^n\}\). What is the matrix associated to the linear transformation \(T:\mathcal{P}_n(\mathbb{F})\to\mathcal{P}_n(\mathbb{F})\) defined by differentiation \(T(f(t)) = f'(t)\)? [Hint. If you don't see immediately how to do it, try doing \(n=2\) and \(n=3\) to get an idea what's going on.]
(d) Continuing with the notation from (c), what is the matrix for the linear transformation \(T:\mathcal{P}_n(\mathbb{F})\to\mathcal{P}_{n+2}(\mathbb{F})\) defined \(T(f(t))=(t^2+1)f(t)\)?
(e) Let \(V\) be the real vector space consisting of all functions of the form \(f(t)=ae^{2t}+bte^{2t}\) with \(a\) and \(b\) in \(\mathbb{R}\). Using the basis \(\{e^{2t},te^{2t}\}\), what is the matrix associated to the linear transformation \(T:V\to V\) defined by \(T(f(t))=f'(t)\)?

Problem # X.5: Compute each of the following matrix products, or explain why the product is not well-defined. \[ \begin{aligned} \hbox{(a)}\quad& \begin{pmatrix} 1&2&3\\ 4&5&6\\ \end{pmatrix} \begin{pmatrix} 3\\-1\\2\\ \end{pmatrix} & \quad\hbox{(b)}\quad& \begin{pmatrix} 1&2&3\\ 4&5&6\\ \end{pmatrix} \begin{pmatrix} 3\\2\\ \end{pmatrix} & \quad\hbox{(c)}\quad& \begin{pmatrix} 1&-1&2\\ 2&1&3\\ -4&2&5\\ \end{pmatrix} \begin{pmatrix} 2&1\\ -1&3\\ 4&-2\\ \end{pmatrix} \\ \hbox{(d)}\quad& \begin{pmatrix} 1&2&3&4\\ \end{pmatrix} \begin{pmatrix} 1\\2\\3\\4\\ \end{pmatrix} & \quad\hbox{(e)}\quad& \begin{pmatrix} 1\\2\\3\\4\\ \end{pmatrix} \begin{pmatrix} 1&2&3&4\\ \end{pmatrix} & \quad\hbox{(f)}\quad& \begin{pmatrix} 3&1&0&0\\ 0&3&1&0\\ 0&0&3&1\\ 0&0&0&3\\ \end{pmatrix} \begin{pmatrix} 3&1&0&0\\ 0&3&1&0\\ 0&0&3&1\\ 0&0&0&3\\ \end{pmatrix} \\ \end{aligned} \]

Problem # X.6: (a) Find linear transformations (or matrices) \(A,B:\mathbb{F}^2\to\mathbb{F}^2\) with the property that \(AB=0\), but \(BA\ne0\).
(b) Find a linear transformation (or matrix) \(A:\mathbb{F}^2\to\mathbb{F}^2\) with the property that \(A\ne 0\), but \(A^2=0\).

Problem # X.7: (a) Give a sequence of swaps that changes the sequence (6,3,1,2,5,4) into the sequence (1,2,3,4,5,6). Use you computation to compute the sign of the permutation (6,3,1,2,5,4).
(b) The sequence (4,3,2,1) can be put into order by the following eight swaps: \[ (4,3,2,1) \to (4,3,1,2) \to (4,1,3,2) \to (4,1,2,3) \to (4,2,1,3) \to (2,4,1,3) \to (2,1,4,3) \to (1,2,4,3) \to (1,2,3,4). \] Find a different way to put the sequence in order using only six swaps. What is the sign of the permuation (4,3,2,1)?

Problem # X.8: Apply the Gram–Schmidt algorithm to the three vectors \[ (1,2,2),\quad (1,1,1)\quad (3,2,1) \quad\text{in}\quad \mathbb{R}^3 \] to create an orthonormal basis. (You can either just keep track of quantities to 4 or 5 decimal places, or express your answer exactly using square roots.)

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