5.11.1.4E: Finite Dimensional Spaces Exercises

Exercises for 1

solutions

2

In each case, find a basis for $V$ that includes the vector $\mathbf{v}$.

1. $V = \mathbb{R}^3$, $\mathbf{v} = (1, -1, 1)$
2. $V = \mathbb{R}^3$, $\mathbf{v} = (0, 1, 1)$
3. $V =\|{M}_{22}$, $\mathbf{v} = \left[ \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right]$
4. $V =\|{P}_{2}$, $\mathbf{v} = x^{2} - x + 1$

2. $\{(0, 1, 1), (1, 0, 0), (0, 1, 0)\}$
3. $\{x^{2} - x + 1, 1, x\}$

In each case, find a basis for $V$ among the given vectors.

1. $V = \mathbb{R}^3$, $\{(1, 1, -1), (2, 0, 1), (-1, 1, -2), (1, 2, 1)\}$
2. $V =\|{P}_{2}$, $\{x^{2} + 3, x + 2, x^{2} - 2x -1, x^{2} + x\}$

2. Any three except $\{x^{2} + 3, x + 2, x^{2} - 2x - 1\}$

In each case, find a basis of $V$ containing $\mathbf{v}$ and $\mathbf{w}$.

1. $V = \mathbb{R}^4$, $\mathbf{v} = (1, -1, 1, -1)$, $\mathbf{w} = (0, 1, 0, 1)$
2. $V = \mathbb{R}^4$, $\mathbf{v} = (0, 0, 1, 1)$, $\mathbf{w} = (1, 1, 1, 1)$
3. $V =\|{M}_{22}$, $\mathbf{v} = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]$, $\mathbf{w} = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]$
4. $V =\|{P}_{3}$, $\mathbf{v} = x^{2} + 1$, $\mathbf{w} = x^{2} + x$

2. Add $(0, 1, 0, 0)$ and $(0, 0, 1, 0)$.
3. Add $1$ and $x^{3}$.

1. If $z$ is not a real number, show that $\{z, z^{2}\}$ is a basis of the real vector space $\mathbb{C}$ of all complex numbers.
2. If $z$ is neither real nor pure imaginary, show that $\{z, \overline{z} \}$ is a basis of $\mathbb{C}$.

2. If $z = a + bi$, then $a \neq 0$ and $b \neq 0$. If $rz + s\overline{z} = 0$, then $(r + s)a = 0$ and $(r - s)b = 0$. This means that $r + s = 0 = r - s$, so $r = s = 0$. Thus $\{z, \overline{z} \}$ is independent; it is a basis because $dim \; \mathbb{C} = 2$.

In each case use Theorem [thm:019633] to decide if $S$ is a basis of $V$.

1. $V =\|{M}_{22}$;
   $S = \left\{ \left[ \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & 0 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & 0 \\ 0 & 1 \end{array} \right] \right\}$

2. $V =\|{P}_{3}$; $S = \{2x^{2}, 1 + x, 3, 1 + x + x^{2} + x^{3}\}$

2. The polynomials in $S$ have distinct degrees.

1. Find a basis of $\mathbf{M}_{22}$ consisting of matrices with the property that $A^{2} = A$.
2. Find a basis of $\mathbf{P}_{3}$ consisting of polynomials whose coefficients sum to $4$. What if they sum to $0$?

2. $\{4, 4x, 4x^{2}, 4x^{3}\}$ is one such basis of $\mathbf{P}_{3}$. However, there is no basis of $\mathbf{P}_{3}$ consisting of polynomials that have the property that their coefficients sum to zero. For if such a basis exists, then every polynomial in $\mathbf{P}_{3}$ would have this property (because sums and scalar multiples of such polynomials have the same property).

If $\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}$ is a basis of $V$, determine which of the following are bases.

1. $\{\mathbf{u} + \mathbf{v}, \mathbf{u} + \mathbf{w}, \mathbf{v} + \mathbf{w}\}$
2. $\{2\mathbf{u} + \mathbf{v} + 3\mathbf{w}, 3\mathbf{u} + \mathbf{v} - \mathbf{w}, \mathbf{u} - 4\mathbf{w}\}$
3. $\{\mathbf{u}, \mathbf{u} + \mathbf{v} + \mathbf{w}\}$
4. $\{\mathbf{u}, \mathbf{u} + \mathbf{w}, \mathbf{u} - \mathbf{w}, \mathbf{v} + \mathbf{w}\}$

2. Not a basis.
3. Not a basis.

1. Can two vectors span $\mathbb{R}^3$? Can they be linearly independent? Explain.
2. Can four vectors span $\mathbb{R}^3$? Can they be linearly independent? Explain.

2. Yes; no.

Show that any nonzero vector in a finite dimensional vector space is part of a basis.

If $A$ is a square matrix, show that $\det A = 0$ if and only if some row is a linear combination of the others.

$\det A = 0$ if and only if $A$ is not invertible; if and only if the rows of $A$ are dependent (Theorem [thm:014205]); if and only if some row is a linear combination of the others (Lemma [lem:019415]).

Let $D$, $I$, and $X$ denote finite, nonempty sets of vectors in a vector space $V$. Assume that $D$ is dependent and $I$ is independent. In each case answer yes or no, and defend your answer.

1. If $X \supseteq D$, must $X$ be dependent?
2. If $X \subseteq D$, must $X$ be dependent?
3. If $X \supseteq I$, must $X$ be independent?
4. If $X \subseteq I$, must $X$ be independent?

2. No. $\{(0, 1), (1, 0)\} \subseteq \{(0, 1), (1, 0), (1, 1)\}$.
3. Yes. See Exercise [ex:6_3_15].

If $U$ and $W$ are subspaces of $V$ and $dim \; U = 2$, show that either $U \subseteq W$ or $dim \;(U \cap W) \leq 1$.

Let $A$ be a nonzero $2 \times 2$ matrix and write $U = \{X \mbox{ in }\|{M}_{22} \mid XA = AX\}$. Show that $dim \; U \geq 2$. [Hint: $I$ and $A$ are in $U$.]

If $U \subseteq \mathbb{R}^2$ is a subspace, show that $U = \{\mathbf{0}\}$, $U = \mathbb{R}^2$, or $U$ is a line through the origin.

Given $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \dots, \mathbf{v}_{k}$, and $\mathbf{v}$, let $U = span \;\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}\}$ and $W = span \;\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}, \mathbf{v}\}$. Show that either $dim \; W = dim \; U$ or $dim \; W = 1 + dim \; U$.

If $\mathbf{v} \in U$ then $W = U$; if $\mathbf{v} \notin U$ then $\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}, \mathbf{v}\}$ is a basis of $W$ by the independent lemma.

Suppose $U$ is a subspace of $\mathbf{P}_{1}$, $U \neq \{0\}$, and $U \neq\|{P}_{1}$. Show that either $U = \mathbb{R}$ or $U = \mathbb{R}(a + x)$ for some $a$ in $\mathbb{R}$.

Let $U$ be a subspace of $V$ and assume $dim \; V = 4$ and $dim \; U = 2$. Does every basis of $V$ result from adding (two) vectors to some basis of $U$? Defend your answer.

Let $U$ and $W$ be subspaces of a vector space $V$.

1. If $dim \; V = 3$, $dim \; U = dim \; W = 2$, and $U \neq W$, show that $dim \;(U \cap W) = 1$.
2. Interpret (a.) geometrically if $V = \mathbb{R}^3$.

2. Two distinct planes through the origin ($U$ and $W$) meet in a line through the origin $(U \cap W)$.

Let $U \subseteq W$ be subspaces of $V$ with $dim \; U = k$ and $dim \; W = m$, where $k < m$. If $k < l < m$, show that a subspace $X$ exists where $U \subseteq X \subseteq W$ and $dim \; X = l$.

Let $B = \{\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\}$ be a maximal independent set in a vector space $V$. That is, no set of more than $n$ vectors $S$ is independent. Show that $B$ is a basis of $V$.

Let $B = \{\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\}$ be a minimal spanning set for a vector space $V$. That is, $V$ cannot be spanned by fewer than $n$ vectors. Show that $B$ is a basis of $V$.

1. Let $p(x)$ and $q(x)$ lie in $\mathbf{P}_{1}$ and suppose that $p(1) \neq 0$, $q(2) \neq 0$, and $p(2) = 0 = q(1)$. Show that $\{p(x), q(x)\}$ is a basis of $\mathbf{P}_{1}$. [Hint: If $rp(x) + sq(x) = 0$, evaluate at $x = 1$, $x = 2$.]
2. Let $B = \{p_{0}(x), p_{1}(x), \dots, p_{n}(x)\}$ be a set of polynomials in $\mathbf{P}_{n}$. Assume that there exist numbers $a_{0}, a_{1}, \dots, a_{n}$ such that $p_{i}(a_{i}) \neq 0$ for each $i$ but $p_{i}(a_{j}) = 0$ if $i$ is different from $j$. Show that $B$ is a basis of $\mathbf{P}_{n}$.

Let $V$ be the set of all infinite sequences $(a_{0}, a_{1}, a_{2}, \dots)$ of real numbers. Define addition and scalar multiplication by

$$(a_{0}, a_{1}, \dots) + (b_{0}, b_{1}, \dots) = (a_{0} + b_{0}, a_{1} + b_{1}, \dots) \nonumber$$

and

$$r(a_{0}, a_{1}, \dots) = (ra_{0}, {ra}_{1}, \dots) \nonumber$$

1. Show that $V$ is a vector space.
2. Show that $V$ is not finite dimensional.
3. [For those with some calculus.] Show that the set of convergent sequences (that is, $\displaystyle \lim_{n \to \infty} a_{n}$ exists) is a subspace, also of infinite dimension.

2. The set $\{(1, 0, 0, 0, \dots), (0, 1, 0, 0, 0, \dots),$
   $(0, 0, 1, 0, 0, \dots), \dots\}$ contains independent subsets of arbitrary size.

Let $A$ be an $n \times n$ matrix of rank $r$. If $U = \{X \mbox{ in }\|{M}_{nn} \mid AX = 0\}$, show that $dim \; U = n(n - r)$. [Hint: Exercise [ex:6_3_34].]

Let $U$ and $W$ be subspaces of $V$.

1. Show that $U + W$ is a subspace of $V$ containing both $U$ and $W$.
2. Show that $span \;\{\mathbf{u}, \mathbf{w}\} = \mathbb{R}\mathbf{u} + \mathbb{R}\mathbf{w}$ for any vectors $\mathbf{u}$ and $\mathbf{w}$.
3. Show that

   $$\begin{aligned} & span \;\{\mathbf{u}_{1}, \dots, \mathbf{u}_{m}, \mathbf{w}_{1}, \dots, \mathbf{w}_{n}\} \\ &= span \;\{\mathbf{u}_{1}, \dots, \mathbf{u}_{m}\} + span \;\{\mathbf{w}_{1}, \dots, \mathbf{w}_{n}\}\end{aligned} \nonumber$$

2. $\mathbb{R}\mathbf{u} + \mathbb{R}\mathbf{w} = \{r\mathbf{u} + s\mathbf{w} \mid r, s \mbox{ in } \mathbb{R}\} = span \;\{\mathbf{u}, \mathbf{w}\}$

If $A$ and $B$ are $m \times n$ matrices, show that $rank \;(A + B) \leq rank \;A + rank \;B$. [Hint: If $U$ and $V$ are the column spaces of $A$ and $B$, respectively, show that the column space of $A + B$ is contained in $U + V$ and that $dim \;(U + V) \leq dim \; U + dim \; V$. (See Theorem [thm:019692].)]