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# 10.E: Exercises for Chapter 10

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## 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

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.