Exercises – Short Day#
Exercise 1: Example of a Change-of-Basis Matrix#
In this exercise we will denote the real vector space \(\mathbb{R}^2\) by \(V\). We are given the following two ordered bases \(\beta\) and \(\gamma\) for \(\mathbb{R}^2\):
and
Question a#
Calculate \(\left[\begin{array}{c} 4\\ 1 \end{array}\right]_\gamma\) and \(\left[\begin{array}{c} 7\\ 2 \end{array}\right]_\gamma\).
Hint
According to Definition 10.2.5 it applies that \(\left[\begin{array}{c} 4\\ 1 \end{array}\right]_\gamma=\left[\begin{array}{c} c_1\\ c_2 \end{array}\right],\) where \(\left[\begin{array}{c} 4\\ 1 \end{array}\right]=c_1\left[\begin{array}{c} 1\\ 2 \end{array}\right]+c_2\left[\begin{array}{c} 2\\ 3 \end{array}\right].\) Now determine \(c_1,c_2\) by solving the corresponding system of linear equations.
Answer
\(\left[\begin{array}{c} 4\\ 1 \end{array}\right]_\gamma=\left[\begin{array}{c} -10\\ 7 \end{array}\right]\) and \(\left[\begin{array}{c} 7\\ 2 \end{array}\right]_\gamma=\left[\begin{array}{c} -17\\ 12 \end{array}\right]\).
Question b#
Use your answers from Question a to determine the change-of-basis matrix \({ }_{\gamma}M_{\beta}\).
Hint
In Theorem 10.2.5 it is described how the matrix \({ }_{\gamma}M_{\beta}\) is defined.
Answer
Exercise 2: More on Change-of-Basis Matrices#
As in Exercise 1 we will denote the real vector space \(\mathbb{R}^2\) by \(V\). The ordered standard basis for this space will be denoted by \(\varepsilon\), while \(\beta\) nd \(\gamma\) are the ordered bases given in Exercise 1. The goal with this exercse is to calculate the change-of-basis matrix \({ }_{\gamma}M_{\beta}\) from Exercise 1 in a different way that sometimes can be easier.
Question a#
Calculate the change-of-basis matrices \({ }_{\varepsilon}M_{\beta}\) and \({ }_{\varepsilon}M_{\gamma}.\)
Answer
and
Note that it generally is easy to determine a change-of-basis matrix of the form \({ }_{\varepsilon}M_{\beta}\) when \(\varepsilon\) is the ordered standard basis for \(\mathbb{F}^n\).
Question b#
Use part (iii) from Lemma 10.3.2 in the textbook along with your answers to Question a to determine the change-of-basis matrix \({ }_{\gamma}M_{\varepsilon}\).
Answer
Question c#
Use part (i) from Lemma 10.3.2 in the textbook along with your answers to Questions a and b to determine the change-of-basis matrix \({ }_{\gamma}M_{\beta}\).
Answer
Exercise 3: The Coordinate Vector of a Vector with respect to Two Ordered Bases#
The notation in this Exercise is the same as in Exercise 2 from Long Day in week 9. In particular, \(W= \{a+bZ+cZ^2 \, \mid \, a,b,c\in \mathbb{C}\} \subset \mathbb{C}[Z]\) is the complex vector space consisting of all polynomials of degree no higher than two. We consider two ordered bases for \(W\), those being \(\beta=(1,Z,Z^2)\) and \(\gamma=(1,1+Z,1+Z+Z^2)\).
Question a#
Determine the change-of-basis matrix \({ }_{\beta}M_\gamma\).
Question b#
We are informed that \(\left[2+5Z+Z^2\right]_\gamma= \left[\begin{array}{c} -3\\ 4\\ 1\\ \end{array}\right]\). Use Theorem 10.3.1 in the textbook to determine \(\left[2+5Z+Z^2\right]_\beta\).
Answer
Question c#
Determine the change-of-basis matrix \({ }_{\gamma}M_\beta\) using part (iii) from Lemma 10.3.2 in the textbook.
Answer
Question d#
We are informed that \(\left[5Z+10Z^2\right]_\beta= \left[\begin{array}{c} 0\\ 5\\ 10\\ \end{array}\right]\). Use Theorem 10.3.1 in the textbook to determine \(\left[5Z+10Z^2\right]_\gamma\).
Answer
Exercise 4: Span of Vectors#
We are given the following four polynomials in \(\mathbb{R}[Z]\): \(p_1(Z)=1+3Z^2\), \(p_2(Z)=-2+Z^2\), \(p_3=5+6Z^2\), and \(p_4(Z)=1+Z^2+Z^5\). We define \(W=\mathrm{Span}(p_1(Z),p_2(Z),p_3(Z),p_4(Z)).\)
Question a#
Determine the ordered basis \(\beta\) for \(W\), which Theorem 10.4.4 in the textbook gives rise to.
Answer
\(\beta=(p_1(Z),p_2(Z),p_(4))\).
Question b#
Show that the polynomial \(p(Z)=Z^5\) is within \(W\), and determine \([p(Z)]_\beta\) where \(\beta\) the ordered basis found in Question a.
Question c#
Some are now claiming that the list \(\gamma=(1,Z^2,Z^5)\) is also an ordered basis for \(W\). Is that claim true?
Answer
Yes.
Question d#
Determine the change-of-coordinates matrices \({ }_\beta M_\gamma\) and \({ }_\gamma M_\beta\). Before you start, take a moment to consider which of these two that is the easiest one to determine.
Answer
and
Exercise 5: Linear Independence#
Let \(V\) be a finite-dimensional vector space and \(\beta\) and ordered basis for \(V\). We are being informed that the set \(([\mathbf{v}_1]_\beta, [\mathbf{v}_2]_\beta, \dots, [\mathbf{v}_n]_\beta)\) for given vectors \(\mathbf{v}_1,\mathbf{v}_2, \cdots, \mathbf{v}_n\) is linearly independent. Show that the set \(([\mathbf{v}_1]_\gamma, [\mathbf{v}_2]_\gamma, \dots, [\mathbf{v}_n]_\gamma)\) is also linearly independent for any other ordered basis \(\gamma\) for \(V\).