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].)]