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Mathematics LibreTexts

10.E: Exercises for Chapter 10

Calculational Exercises

1. Consider \(\mathbb{R}^3 \) with two orthonormal bases: the canonical basis \(e = (e_1 , e_2 , e_3 )\) and the basis \(f = (f_1 , f_2 , f_3)\), where

                            \[ f_1 = \frac{1}{\sqrt{3}}(1,1,1),  f_2 = \frac{1}{\sqrt{6}}(1,-2,1), f_3 = \frac{1}{\sqrt{2}}(1,0,-1) \]
 

Find the matrix, \(S\), of the change of basis transformation such that
                                                                             \[ [v]_f = S[v]_e, ~\rm{for~ all}~ v \in \mathbb{R}^3 ,\] 

where \([v]_b\) denotes the column vector of \(v\) with respect to the basis \(b\).

2. Let \(v \in \mathbb{C}^4\) be the vector given by \(v = (1, i, −1, −i)\). Find the matrix (with respect to the canonical basis on \(\mathbb{C}^4\) ) of the orthogonal projection \(P \in \cal{L}(\mathbb{C}^4)\) such that

                                                                                                         \[null(P ) = {v}^\perp.\]

3. Let \(U\) be the subspace of \(\mathbb{R}^3\) that coincides with the plane through the origin that is perpendicular to the vector \(n = (1, 1, 1) \in \mathbb{R}^3.\)

(a) Find an orthonormal basis for \(U\).

(b) Find the matrix (with respect to the canonical basis on \(\mathbb{R}^3\)) of the orthogonal projection \(P \in \cal{L}(\mathbb{R}^3\)) onto \(U\), i.e., such that \(range(P ) = U\).

4. Let \(V = \mathbb{C}^4\) with its standard inner product. For \( \theta \in \mathbb{R}\), let

                         \[ v_\theta = \left( \begin{array}{c} 1 \\ e^{i\theta} \\ e^{2i\theta} \\ e^{3i\theta} \end{array} \right) \in \mathbb{C}^4.\]

Find the canonical matrix of the orthogonal projection onto the subspace \({v_\theta }^\perp\).

Proof-Writing Exercises

1. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with dimension \(n \in \mathbb{Z}_+ \), and suppose that \(b = (v_1 , v_2 , \ldots , v_n) \) is a basis for \(V\) . Prove that the coordinate vectors \([v_1 ]_b, [v_2 ]_b, \ldots, [v_n ]_b\) with respect to \(b\) form a basis for \(\mathbb{F}^n.\)

2. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\), and suppose that \(T \in \cal{L}(V)\) is a linear operator having the following property: Given any two bases \(b\) and \(c\) for \(V\) , the matrix \(M(T, b)\) for \(T\) with respect to \(b\) is the same as the matrix \(M(T, c)\) for \(T\) with respect to \(c\). Prove that there exists a scalar \(\alpha \in \mathbb{F}\) such that \(T = \alpha I_V\), where \(I_V\) denotes the identity map on \(V\).

Contributors

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