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# 5.E: Exercises for Chapter 5

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#### Calculational Exercises

1. Show that the vectors $$v_1 = (1, 1, 1), v_2 = (1, 2, 3)$$, and $$v_3 = (2, −1, 1)$$ are linearly independent in $$\mathbb{R}^3$$. Write $$v = (1, −2, 5)$$ as a linear combination of $$v_1 , v_2$$, and $$v_3$$.

2. Consider the complex vector space $$V = \mathbb{C}^3$$ and the list $$(v_1 , v_2 , v_3 )$$ of vectors in $$V$$ , where

$v_1 = (i, 0, 0),~ v_2 = (i, 1, 0),~ v_3 = (i, i, −1).$

(a) Prove that $$span(v_1 , v-2 , v_3 ) = V.$$
(b) Prove or disprove: $$(v_1 , v_2 , v_3)$$ is a basis for $$V.$$

3. Determine the dimension of each of the following subspaces of $$\mathbb{F}^4$$ .
(a) $$\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = 0\}.$$
(b) $$\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 \}.$$
(c) $$\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 \}.$$
(d) $$\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 , x_3 + x_4 = 2x_1 \}.$$
(e) $$\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_1 = x_2 = x_3 = x_4 \}.$$

4. Determine the value of $$\lambda \in \mathbb{R}$$ for which each list of vectors is linear dependent.
(a)  $$((\lambda, −1, −1), (−1, \lambda, −1), (−1, −1, \lambda))$$ as a subset of $$\mathbb{R}^3.$$
(b)  $$sin2 (x), cos(2x), \lambda$$ as a subset of $$\cal{C}(\mathbb{R}).$$

5. Consider the real vector space $$V = \mathbb{R}^4.$$ For each of the following ﬁve statements, provide either a proof or a counterexample.
(a) $$dim V = 4.$$
(b) $$span((1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1)) = V.$$
(c) The list $$((1, −1, 0, 0), (0, 1, −1, 0), (0, 0, 1, −1), (−1, 0, 0, 1))$$ is linearly independent.
(d) Every list of four vectors $$v_1 , \ldots, v_4 \in V$$ , such that $$span(v_1 , \ldots, v_4 ) = V$$ , is linearly independent.
(e) Let $$v_1$$ and $$v_2$$ be two linearly independent vectors in $$V$$ . Then, there exist vectors $$u, w \in V$$ , such that $$(v_1 , v_2 , u, w)$$ is a basis for $$V.$$

#### Proof-Writing Exercises

1.  Let $$V$$ be a vector space over $$\mathbb{F}$$ and define $$U = span(u_1, u_2, \ldots ,u_n)$$, where for each
$$i = 1, \ldots ,n, u_i \in V$$. Now suppose $$v \in U$$. Prove

$U = span(v, u_1, u_2, \ldots ,u_n) .$

2. Let $$V$$ be a vector space over $$\mathbb{F}$$, and suppose that the list $$(v_1 , v_2 , . . . , v_n )$$ of vectors spans $$V$$ , where each $$v_i \in V$$ . Prove that the list

$(v_1 − v_2 , v_2 − v_3 , v_3 − v_4 , \ldots , v_{n−2} − v_{n−1} , v_{n−1} − v_n , v_n )$

also spans $$V.$$

3. Let $$V$$ be a vector space over $$\mathbb{F}$$, and suppose that $$(v_1 , v_2 , \ldots, v_n)$$ is a linearly independent list of vectors in $$V$$ . Given any $$w \in V$$ such that

$(v_1 + w, v_2 + w, \ldots , v_n + w)$

is a linearly dependent list of vectors in $$V$$ , prove that $$w \in span(v_1 , v_2 , \ldots, v_n).$$

4. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$ with $$dim(V ) = n$$ for some $$n \in \mathbb{Z}_+$$. Prove that there are $$n$$ one-dimensional subspaces $$U_1 , U_2 , \ldots , U_n$$ of $$V$$ such that

$V = U_1 \oplus U_2 \oplus \cdots \oplus U_n .$

5. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$, and suppose that $$U$$ is a subspace of $$V$$ for which $$dim(U) = dim(V ).$$ Prove that $$U = V.$$

6. Let $$\mathbb{F}_m [z]$$ denote the vector space of all polynomials with degree less than or equal to $$m \in \mathbb{Z}_+$$ and having coeﬃcient over $$\mathbb{F}$$, and suppose that$$p_0 , p_1 , \ldots , p_m \in \mathbb{F}_m [z]$$ satisfy $$p_j (2) = 0$$. Prove that $$(p_0 , p_1 , \ldots , p_m )$$ is a linearly dependent list of vectors in $$\mathbb{F}_m [z].$$

7. Let $$U$$ and $$V$$ be ﬁve-dimensional subspaces of $$\mathbb{R}^9$$ . Prove that $$U \cap V = \{0\}.$$

8. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and suppose that $$U_1 , U_2 , \ldots, U_m$$ are any $$m$$ subspaces of $$V$$ . Prove that

$dim(U_1 + U_2 + \cdots + U_m ) \leq dim(U_1 ) + dim(U_2 ) + \cdots + dim(U_m ).$

### Contributors

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