5.11.1.3E: Linear Independence and Dimension Exercises

Exercise \(\PageIndex{1}\)

Show that each of the following sets of vectors is independent.

1. \(\{1 + x, 1 - x, x + x^{2}\}\) in \(\mathbf{P}_{2}\)
2. \(\{x^{2}, x + 1, 1 - x - x^{2}\}\) in \(\mathbf{P}_{2}\)
3. \(\left\{ \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 1 & 0 \end{array} \right] , \left[ \begin{array}{rr} 0 & 0 \\ 1 & -1 \end{array} \right] ,\ \left[ \begin{array}{rr} 0 & 1 \\ 0 & 1 \end{array} \right] \right\}\) in \(\mathbf{M}_{22}\)
4. \(\left\{ \left[ \begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array} \right] , \left[ \begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] ,\ \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \right\}\) in \(\mathbf{M}_{22}\)

Answer

2. If \(ax^{2} + b(x + 1) + c(1 - x - x^{2}) = 0\), then \(a + c = 0\), \(b - c = 0\), \(b + c = 0\), so \(a = b = c = 0\).
4. If \(a \left[ \begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array} \right] + b \left[ \begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array} \right] + c \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] + d \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\), then \(a + c + d = 0\), \(a + b + d = 0\), \(a + b + c = 0\), and \(b + c + d = 0\), so \(a = b = c = d = 0\).

Exercise \(\PageIndex{2}\)

Which of the following subsets of \(V\) are independent?

1. \(V =\|{P}_{2}\); \(\{x^{2} + 1, x + 1, x\}\)
2. \(V =\|{P}_{2}\); \(\{x^{2} - x + 3, 2x^{2} + x + 5, x^{2} + 5x + 1\}\)
3. \(V =\|{M}_{22}\); \(\left\{ \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] \right\}\)
4. \(V =\|{M}_{22}\); \(\small{\left\{ \left[ \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right] , \left[ \begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array} \right] \right\}}\)
5. \(V =\|{F}[1, 2]\); \(\left\{\frac{1}{x}, \frac{1}{x^2}, \frac{1}{x^3} \right\}\)
6. \(V =\|{F}[0, 1]\); \(\left\{\frac{1}{x^2 + x - 6}, \frac{1}{x^2 - 5x + 6}, \frac{1}{x^2 -9} \right\}\)

Answer

2. \(3(x^{2} - x + 3) - 2(2x^{2} + x + 5) + (x^{2} + 5x + 1) = 0\)
4. \(2 \left[ \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right] + \left[ \begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array} \right] + \left[ \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right] = \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\)
6. \(\frac{5}{x^2 + x - 6} + \frac{1}{x^2 - 5x + 6} - \frac{6}{x^2 - 9} = 0\)

Exercise \(\PageIndex{3}\)

Which of the following are independent in \(\mathbf{F}[0, 2\pi]\)?

1. \(\{\sin^{2} x, \cos^{2} x\}\)
2. \(\{1, \sin^{2} x, \cos^{2} x\}\)
3. \(\{x, \sin^{2} x, \cos^{2} x\}\)

Answer

2. Dependent: \(1 - \sin^{2} x - \cos^{2} x = 0\)

Exercise \(\PageIndex{4}\)

Find all values of \(a\) such that the following are independent in \(\mathbb{R}^3\).

1. \(\{(1, -1, 0), (a, 1, 0), (0, 2, 3)\}\)
2. \(\{(2, a, 1), (1, 0, 1), (0, 1, 3)\}\)

Answer

2. \(x \neq -\frac{1}{3}\)

Exercise \(\PageIndex{5}\)

Show that the following are bases of the space \(V\) indicated.

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

Answer

2. If \(r(-1, 1, 1) + s(1, -1, 1) + t(1, 1, -1) = (0, 0, 0)\), then \(-r + s + t = 0\), \(r - s + t = 0\), and \(r - s - t = 0\), and this implies that \(r = s = t = 0\). This proves independence. To prove that they \(span \; \mathbb{R}^3\), observe that \((0, 0, 1) = \frac{1}{2}[(-1, 1, 1) + (1, -1, 1)]\) so \((0, 0, 1)\) lies in \(span \;\{(-1, 1, 1), (1, -1, 1), (1, 1, -1)\}\). The proof is similar for \((0, 1, 0)\) and \((1, 0, 0)\).
4. If \(r(1 + x) + s(x + x^{2}) + t(x^{2} + x^{3}) + ux^{3} = 0\), then \(r = 0\), \(r + s = 0\), \(s + t = 0\), and \(t + u = 0\), so \(r = s = t = u = 0\). This proves independence. To show that they \(span \;\|{P}_{3}\), observe that \(x^{2} = (x^{2} + x^{3}) - x^{3}\), \(x = (x + x^{2}) - x^{2}\), and \(1 = (1 + x) - x\), so \(\{1, x, x^{2}, x^{3}\} \subseteq span \;\{1 + x, x + x^{2}, x^{2} + x^{3}, x^{3}\}\).

Exercise \(\PageIndex{6}\)

Exhibit a basis and calculate the dimension of each of the following subspaces of \(\mathbf{P}_{2}\).

1. \(\{a(1 + x) + b(x + x^{2}) \mid a \mbox{ and } b \mbox{ in } \mathbb{R}\}\)
2. \(\{a + b(x + x^{2}) \mid a \mbox{ and } b \mbox{ in } \mathbb{R}\}\)
3. \(\{p(x) \mid p(1) = 0\}\)
4. \(\{p(x) \mid p(x) = p(-x)\}\)

Answer

1. \(\{1, x + x^{2}\}\); dimension \(= 2\)
4. \(\{1, x^{2}\}\); dimension \(= 2\)

Exercise \(\PageIndex{7}\)

Exhibit a basis and calculate the dimension of each of the following subspaces of \(\mathbf{M}_{22}\).

1. \(\{A \mid A^{T} = -A\}\)
2. \(\left\{ A\ \middle|\ A \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right] A \right\}\)
3. \(\left \{ A\ \middle|\ A \left[ \begin{array}{rr} 1 & 0 \\ -1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right] \right \}\)
4. \(\left\{ A\ \middle|\ A \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 0 & 1 \\ -1 & 1 \end{array} \right] A \right\}\)

Answer

1. \(\left\{ \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] \right\}\); dimension \(= 2\)
4. \(\left\{ \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right] \right\}\); dimension \(= 2\)

Exercise \(\PageIndex{8}\)

Let \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right]\) and define \(U = \{X \mid X \in\|{M}_{22} \mbox{ and } AX = X\}\).

1. Find a basis of \(U\) containing \(A\).
2. Find a basis of \(U\) not containing \(A\).

Answer

1. \(\left\{ \left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right] , \left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right] \right\}\)

Exercise \(\PageIndex{9}\)

Show that the set \(\mathbb{C}\) of all complex numbers is a vector space with the usual operations, and find its dimension.

1. Let \(V\) denote the set of all \(2 \times 2\) matrices with equal column sums. Show that \(V\) is a subspace of \(\mathbf{M}_{22}\), and compute \(dim \; V\).
4. Repeat part (a) for \(3 \times 3\) matrices.
6. Repeat part (a) for \(n \times n\) matrices.

Answer

1. \(dim \; V = 7\)

Exercise \(\PageIndex{10}\)

1. Let \(V\) denote the set of all \(2 \times 2\) matrices with equal column sums. Show that \(V\) is a subspace of \(\mathbf{M}_{22}\), and compute \(\operatorname{dim} V\).
2. Repeat part (a) for \(3 \times 3\) matrices.
3. Repeat part (a) for \(n \times n\) matrices.

Exercise \(\PageIndex{11}\)

1. Let \(V = \{(x^{2} + x + 1)p(x) \mid p(x) \mbox{ in }\|{P}_{2}\}\). Show that \(V\) is a subspace of \(\mathbf{P}_{4}\) and find \(dim \; V\). [Hint: If \(f(x)g(x) = 0\) in \(\mathbf{P}\), then \(f(x) = 0\) or \(g(x) = 0\).]
4. Repeat with \(V = \{(x^{2} - x)p(x) \mid p(x) \mbox{ in }\|{P}_{3}\}\), a subset of \(\mathbf{P}_{5}\).
6. Generalize.

Answer

1. \(\{x^{2} - x, x(x^{2} - x), x^{2}(x^{2} - x), x^{3}(x^{2} - x)\}\); \(dim \; V = 4\)

Exercise \(\PageIndex{12}\)

In each case, either prove the assertion or give an example showing that it is false.

1. Every set of four nonzero polynomials in \(\mathbf{P}_{3}\) is a basis.
2. \(\mathbf{P}_{2}\) has a basis of polynomials \(f(x)\) such that \(f(0) = 0\).
3. \(\mathbf{P}_{2}\) has a basis of polynomials \(f(x)\) such that \(f(0) = 1\).
4. Every basis of \(\mathbf{M}_{22}\) contains a noninvertible matrix.
5. No independent subset of \(\mathbf{M}_{22}\) contains a matrix \(A\) with \(A^{2} = 0\).
6. If \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent then, \(a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}\) for some \(a\), \(b\), \(c\).
7. \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent if \(a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}\) for some \(a\), \(b\), \(c\).
8. If \(\{\mathbf{u}, \mathbf{v}\}\) is independent, so is \(\{\mathbf{u}, \mathbf{u} + \mathbf{v}\}\).
9. If \(\{\mathbf{u}, \mathbf{v}\}\) is independent, so is \(\{\mathbf{u}, \mathbf{v}, \mathbf{u} + \mathbf{v}\}\).
10. If \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent, so is \(\{\mathbf{u}, \mathbf{v}\}\).
11. If \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent, so is \(\{\mathbf{u} + \mathbf{w}, \mathbf{v} + \mathbf{w}\}\).
12. If \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent, so is \(\{\mathbf{u} + \mathbf{v} + \mathbf{w}\}\).
13. If \(\mathbf{u} \neq \mathbf{0}\) and \(\mathbf{v} \neq \mathbf{0}\) then \(\{\mathbf{u}, \mathbf{v}\}\) is dependent if and only if one is a scalar multiple of the other.
14. If \(dim \; V = n\), then no set of more than \(n\) vectors can be independent.
15. If \(dim \; V = n\), then no set of fewer than \(n\) vectors can span \(V\).

Answer

1. No. Any linear combination \(f\) of such polynomials has \(f(0) = 0\).
4. No. \(\left\{ \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] , \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right] , \left[ \begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array} \right] \right\}\); consists of invertible matrices.
6. Yes. \(0\mathbf{u} + 0\mathbf{v} + 0\mathbf{w} = \mathbf{0}\) for every set \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\).
8. Yes. \(s\mathbf{u} + t(\mathbf{u} + \mathbf{v}) = \mathbf{0}\) gives \((s + t)\mathbf{u} + t\mathbf{v} = \mathbf{0}\), whence \(s + t = 0 = t\).
10. Yes. If \(r\mathbf{u} + s\mathbf{v} = \mathbf{0}\), then \(r\mathbf{u} + s\mathbf{v} + 0\mathbf{w} = \mathbf{0}\), so \(r = 0 = s\).
12. Yes. \(\mathbf{u} + \mathbf{v} + \mathbf{w} \neq \mathbf{0}\) because \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent.
14. Yes. If \(I\) is independent, then \(|I| \leq n\) by the fundamental theorem because any basis spans \(V\).

Exercise \(\PageIndex{13}\)

Let \(A \neq 0\) and \(B \neq 0\) be \(n \times n\) matrices, and assume that \(A\) is symmetric and \(B\) is skew-symmetric (that is, \(B^{T} = -B\)). Show that \(\{A, B\}\) is independent.

Exercise \(\PageIndex{14}\)

Show that every set of vectors containing a dependent set is again dependent.

Exercise \(\PageIndex{15}\)

Show that every nonempty subset of an independent set of vectors is again independent.

Answer

If a linear combination of the subset vanishes, it is a linear combination of the vectors in the larger set (coefficients outside the subset are zero) so it is trivial.

Exercise \(\PageIndex{16}\)

Let \(f\) and \(g\) be functions on \([a, b]\), and assume that \(f(a) = 1 = g(b)\) and \(f(b) = 0 = g(a)\). Show that \(\{f, g\}\) is independent in \(\mathbf{F}[a, b]\).

Exercise \(\PageIndex{17}\)

Let \(\{A_{1}, A_{2}, \dots, A_{k}\}\) be independent in \(\mathbf{M}_{mn}\), and suppose that \(U\) and \(V\) are invertible matrices of size \(m \times m\) and \(n \times n\), respectively. Show that \(\{UA_{1}V, UA_{2}V, \dots, UA_{k}V\}\) is independent.

Exercise \(\PageIndex{18}\)

Show that \(\{\mathbf{v}, \mathbf{w}\}\) is independent if and only if neither \(\mathbf{v}\) nor \(\mathbf{w}\) is a scalar multiple of the other.

Exercise \(\PageIndex{19}\)

Assume that \(\{\mathbf{u}, \mathbf{v}\}\) is independent in a vector space \(V\). Write \(\mathbf{u}^\prime = a\mathbf{u} + b\mathbf{v}\) and \(\mathbf{v}^\prime = c\mathbf{u} + d\mathbf{v}\), where \(a\), \(b\), \(c\), and \(d\) are numbers. Show that \(\{\mathbf{u}^\prime, \mathbf{v}^\prime\}\) is independent if and only if the matrix \(\left[ \begin{array}{rr} a & c \\ b & d \end{array} \right]\) is invertible. [Hint: Theorem 2.4.5.]

Answer

Because \(\{\mathbf{u}, \mathbf{v}\}\) is linearly independent, \(s\mathbf{u}^\prime + t\mathbf{v}^\prime = \mathbf{0}\) is equivalent to \(\left[ \begin{array}{rr} a & c \\ b & d \end{array} \right] \left[ \begin{array}{r} s \\ t \end{array} \right] = \left[ \begin{array}{r} 0 \\ 0 \end{array} \right]\). Now apply Theorem 2.4.5.

Exercise \(\PageIndex{20}\)

If \(\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}\}\) is independent and \(\mathbf{w}\) is not in \(span \;\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}\}\), show that:

1. \(\{\mathbf{w}, \mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}\}\) is independent.
2. \(\{\mathbf{v}_{1} + \mathbf{w}, \mathbf{v}_{2} + \mathbf{w}, \dots, \mathbf{v}_{k} + \mathbf{w}\}\) is independent.

Exercise \(\PageIndex{21}\)

If \(\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k}\}\) is independent, show that \(\{\mathbf{v}_{1}, \mathbf{v}_{1} + \mathbf{v}_{2}, \dots, \mathbf{v}_{1} + \mathbf{v}_{2} + \dots + \mathbf{v}_{k}\}\) is also independent.

Exercise \(\PageIndex{22}\)

Prove Example 6.3.12.

Exercise \(\PageIndex{23}\)

Let \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z}\}\) be independent. Which of the following are dependent?

1. \(\{\mathbf{u} - \mathbf{v}, \mathbf{v} - \mathbf{w}, \mathbf{w} - \mathbf{u}\}\)
2. \(\{\mathbf{u} + \mathbf{v}, \mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{u}\}\)
3. \(\{\mathbf{u} - \mathbf{v}, \mathbf{v} - \mathbf{w}, \mathbf{w} - \mathbf{z}, \mathbf{z} - \mathbf{u}\}\)
4. \(\{\mathbf{u} + \mathbf{v}, \mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{z}, \mathbf{z} + \mathbf{u}\}\)

Answer

2. Independent.
4. Dependent. For example, \((\mathbf{u} + \mathbf{v}) - (\mathbf{v} + \mathbf{w}) + (\mathbf{w} + \mathbf{z}) - (\mathbf{z} + \mathbf{u}) = \mathbf{0}\).

Exercise \(\PageIndex{24}\)

Let \(U\) and \(W\) be subspaces of \(V\) with bases \(\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\}\) and \(\{\mathbf{w}_{1}, \mathbf{w}_{2}\}\) respectively. If \(U\) and \(W\) have only the zero vector in common, show that \(\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{w}_{1}, \mathbf{w}_{2}\}\) is independent.

Exercise \(\PageIndex{25}\)

Let \(\{p, q\}\) be independent polynomials. Show that \(\{p, q, pq\}\) is independent if and only if \(\text{deg} p \geq 1\) and \(\text{deg} q \geq 1\).

Exercise \(\PageIndex{26}\)

If \(z\) is a complex number, show that \(\{z, z^{2}\}\) is independent if and only if \(z\) is not real.

Answer

If \(z\) is not real and \(az + bz^{2} = 0\), then \(a + bz = 0 (z \neq 0)\). Hence if \(b \neq 0\), then \(z = -ab^{-1}\) is real. So \(b = 0\), and so \(a = 0\). Conversely, if \(z\) is real, say \(z = a\), then \((-a)z + 1z^{2} = 0\), contrary to the independence of \(\{z, z^{2}\}\).

Exercise \(\PageIndex{27}\)

Let \(B = \{A_{1}, A_{2}, \dots, A_{n}\} \subseteq\|{M}_{mn}\), and write \(B^\prime = \{A_1^T, A_2^T, \dots, A_n^T\} \subseteq\|{M}_{nm}\). Show that:

1. \(B\) is independent if and only if \(B^\prime\) is independent.
2. \(B\) spans \(\mathbf{M}_{mn}\) if and only if \(B^\prime\) spans \(\mathbf{M}_{nm}\).

Exercise \(\PageIndex{28}\)

If \(V =\|{F}[a, b]\) as in Example 6.1.7, show that the set of constant functions is a subspace of dimension \(1\) (\(f\) is constant if there is a number \(c\) such that \(f(x) = c\) for all \(x\)).

Exercise \(\PageIndex{29}\)

1. If \(U\) is an invertible \(n \times n\) matrix and \(\{A_{1}, A_{2}, \dots, A_{mn}\}\) is a basis of \(\mathbf{M}_{mn}\), show that \(\{A_{1}U, A_{2}U, \dots, A_{mn}U\}\) is also a basis.
2. Show that part (a) fails if \(U\) is not invertible. [Hint: Theorem 2.4.5.]

Answer

2. If \(U\mathbf{x} = \mathbf{0}\), \(\mathbf{x} \neq \mathbf{0}\) in \(\mathbb{R}^n\), then \(R\mathbf{x} = \mathbf{0}\) where \(R \neq 0\) is row 1 of \(U\). If \(B \in\|{M}_{mn}\) has each row equal to \(R\), then \(B\mathbf{x} \neq \mathbf{0}\). But if \(B = \sum r_{i}A_{i}U\), then \(B\mathbf{x} = \sum r_{i}A_{i}U\mathbf{x} = \mathbf{0}\). So \(\{A_{i}U\}\) cannot span \(\mathbf{M}_{mn}\).

Exercise \(\PageIndex{30}\)

Show that \(\{(a, b), (a_{1}, b_{1})\}\) is a basis of \(\mathbb{R}^2\) if and only if \(\{a + bx, a_{1} + b_{1}x\}\) is a basis of \(\mathbf{P}_{1}\).

Exercise \(\PageIndex{31}\)

Find the dimension of the subspace \(span \;\{1, \sin^{2} \theta, \cos 2\theta\}\) of \(\mathbf{F}[0, 2\pi]\).

Exercise \(\PageIndex{32}\)

Show that \(\mathbf{F}[0, 1]\) is not finite dimensional.

Exercise \(\PageIndex{33}\)

If \(U\) and \(W\) are subspaces of \(V\), define their intersection \(U \cap W\) as follows:

\[U \cap W = \{\mathbf{v} \mid \mathbf{v} \mbox{ is in both } U \mbox{ and } W\} \nonumber\]

1. Show that \(U \cap W\) is a subspace contained in \(U\) and \(W\).
2. Show that \(U \cap W = \{\mathbf{0}\}\) if and only if \(\{\mathbf{u}, \mathbf{w}\}\) is independent for any nonzero vectors \(\mathbf{u}\) in \(U\) and \(\mathbf{w}\) in \(W\).
3. If \(B\) and \(D\) are bases of \(U\) and \(W\), and if \(U \cap W = \{\mathbf{0}\}\), show that \(B \cup D = \{\mathbf{v} \mid \mathbf{v} \mbox{ is in } B \mbox{ or } D\}\) is independent.

Answer

2. If \(U \cap W = 0\) and \(r\mathbf{u} + s\mathbf{w} = \mathbf{0}\), then \(r\mathbf{u} = -s\mathbf{w}\) is in \(U \cap W\), so \(r\mathbf{u} = \mathbf{0} = s\mathbf{w}\). Hence \(r = 0 = s\) because \(\mathbf{u} \neq \mathbf{0} \neq \mathbf{w}\). Conversely, if \(\mathbf{v} \neq \mathbf{0}\) lies in \(U \cap W\), then \(1\mathbf{v} + (-1)\mathbf{v} = \mathbf{0}\), contrary to hypothesis.

Exercise \(\PageIndex{34}\)

If \(U\) and \(W\) are vector spaces, let \(V = \{(\mathbf{u}, \mathbf{w}) \mid \mathbf{u} \mbox{ in } U \mbox{ and } \mathbf{w} \mbox{ in } W\}\).

1. Show that \(V\) is a vector space if \((\mathbf{u}, \mathbf{w}) + (\mathbf{u}_{1}, \mathbf{w}_{1}) = (\mathbf{u} + \mathbf{u}_{1}, \mathbf{w} + \mathbf{w}_{1})\) and \(a(\mathbf{u}, \mathbf{w}) = (a\mathbf{u}, a\mathbf{w})\).
2. If \(dim \; U = m\) and \(dim \; W = n\), show that \(dim \; V = m + n\).
3. If \(V_{1}, \dots, V_{m}\) are vector spaces, let \[\begin{aligned} V &= V_{1} \times \dots \times V_{m} \\ &= \{(\mathbf{v}_{1}, \dots, \mathbf{v}_{m}) \mid \mathbf{v}_{i} \in V_{i} \mbox{ for each } i\}\end{aligned} \nonumber \]

Exercise \(\PageIndex{35}\)

Let \(\mathbf{D}_{n}\) denote the set of all functions \(f\) from the set \(\{1, 2, \dots, n\}\) to \(\mathbb{R}\).

1. Show that \(\mathbf{D}_{n}\) is a vector space with pointwise addition and scalar multiplication.
2. Show that \(\{S_{1}, S_{2}, \dots, S_{n}\}\) is a basis of \(\mathbf{D}_{n}\) where, for each \(k = 1, 2, \dots, n\), the function \(S_{k}\) is defined by \(S_{k}(k) = 1\), whereas \(S_{k}(j) = 0\) if \(j \neq k\).

Exercise \(\PageIndex{36}\)

A polynomial \(p(x)\) is called even if \(p(-x) = p(x)\) and odd if \(p(-x) = -p(x)\). Let \(E_{n}\) and \(O_{n}\) denote the sets of even and odd polynomials in \(\mathbf{P}_{n}\).

1. Show that \(E_{n}\) is a subspace of \(\mathbf{P}_{n}\) and find \(dim \; E_{n}\).
2. Show that \(O_{n}\) is a subspace of \(\mathbf{P}_{n}\) and find \(dim \; O_{n}\).

Answer

2. \(dim \; O_n = \frac{n}{2}\) if \(n\) is even and \(dim \; O_n = \frac{n + 1}{2}\) if \(n\) is odd.

Exercise \(\PageIndex{37}\)

Let \(\{\mathbf{v}_{1}, \dots, \mathbf{v}_{n}\}\) be independent in a vector space \(V\), and let \(A\) be an \(n \times n\) matrix. Define \(\mathbf{u}_{1}, \dots, \mathbf{u}_{n}\) by

\[\left[ \begin{array}{c} \mathbf{u}_1 \\ \vdots \\ \mathbf{u}_n \end{array} \right] = A \left[ \begin{array}{c} \mathbf{v}_1 \\ \vdots \\ \mathbf{v}_n \end{array} \right] \nonumber \] Show that \(\{\mathbf{u}_{1}, \dots, \mathbf{u}_{n}\}\) is independent if and only if \(A\) is invertible.