5.E: Exercises for Chapter 5

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

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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