5.11.1.3E: Linear Independence and Dimension Exercises

Exercises for 1

solutions

2

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 \(\mathbf{M}_{22}\)

2. If \(ax^{2} + b(x + 1) + c(1 - x - x^{2}) = 0\), then \(a + c = 0\), \(b - c = 0\), \(b + c = 0\), so \(a = b = c = 0\).
3. 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 \(a = b = c = d = 0\).

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. \(V =\|{F}[0, 1]\); \(\left\{\frac{1}{x^2 + x - 6}, \frac{1}{x^2 - 5x + 6}, \frac{1}{x^2 -9} \right\}\)

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

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

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

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

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

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

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 \((1, 0, 0)\).
3. 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 \(\{1, x, x^{2}, x^{3}\} \subseteq span \;\{1 + x, x + x^{2}, x^{2} + x^{3}, x^{3}\}\).

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. \(\{p(x) \mid p(x) = p(-x)\}\)

2. \(\{1, x + x^{2}\}\); dimension \(= 2\)
3. \(\{1, x^{2}\}\); dimension \(= 2\)

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. \(\left\{ A\ \middle|\ A \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 0 & 1 \\ -1 & 1 \end{array} \right] A \right\}\)

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

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 \(A\).

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

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 \(dim \; V\).
2. Repeat part (a) for \(3 \times 3\) matrices.
3. Repeat part (a) for \(n \times n\) matrices.

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

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 \(g(x) = 0\).]
2. Repeat with \(V = \{(x^{2} - x)p(x) \mid p(x) \mbox{ in }\|{P}_{3}\}\), a subset of \(\mathbf{P}_{5}\).
3. Generalize.

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

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 \(V\).

2. No. Any linear combination \(f\) of such polynomials has \(f(0) = 0\).

3. 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.

4. Yes. \(0\mathbf{u} + 0\mathbf{v} + 0\mathbf{w} = \mathbf{0}\) for every set \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\).
5. Yes. \(s\mathbf{u} + t(\mathbf{u} + \mathbf{v}) = \mathbf{0}\) gives \((s + t)\mathbf{u} + t\mathbf{v} = \mathbf{0}\), whence \(s + t = 0 = t\).
6. Yes. If \(r\mathbf{u} + s\mathbf{v} = \mathbf{0}\), then \(r\mathbf{u} + s\mathbf{v} + 0\mathbf{w} = \mathbf{0}\), so \(r = 0 = s\).
7. Yes. \(\mathbf{u} + \mathbf{v} + \mathbf{w} \neq \mathbf{0}\) because \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is independent.
8. Yes. If \(I\) is independent, then \(|I| \leq n\) by the fundamental theorem because any basis spans \(V\).

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.

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

[ex:6_3_15] Show that every nonempty subset of an independent set of vectors is again independent.

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.

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 \(\mathbf{F}[a, b]\).

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.

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.

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 [thm:004553].]

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 [thm:004553].

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.

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.

[ex:6_3_22] Prove Example [exa:018943].

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. \(\{\mathbf{u} + \mathbf{v}, \mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{z}, \mathbf{z} + \mathbf{u}\}\)

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

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.

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

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

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 \(\{z, z^{2}\}\).

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 \(\mathbf{M}_{nm}\).

If \(V =\|{F}[a, b]\) as in Example [exa:017760], 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 \(x\)).

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 [thm:004553].]

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 \(\mathbf{M}_{mn}\).

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 \(\mathbf{P}_{1}\).

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

[ex:6_3_32] Show that \(\mathbf{F}[0, 1]\) is not finite dimensional.

[ex:ex6_3_33] If \(U\) and \(W\) are subspaces of \(V\), define their intersection \(U \cap W\) as follows:

\(U \cap W = \{\mathbf{v} \mid \mathbf{v} \mbox{ is in both } U \mbox{ and } W\}\)

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.

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.

[ex:6_3_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

   \[\begin{aligned} V &= V_{1} \times \dots \times V_{m} \\ &= \{(\mathbf{v}_{1}, \dots, \mathbf{v}_{m}) \mid \mathbf{v}_{i} \in V_{i} \mbox{ for each } i\}\end{aligned} \nonumber \]

[ex:ex6_3_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 \(j \neq k\).

[ex:ex6_3_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 \(dim \; O_{n}\).

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

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

\[\left[ \begin{array}{c} \mathbf{u}_1 \\ \vdots \\ \mathbf{u}_n \end{array} \right] = A \left[ \begin{array}{c} \mathbf{v}_1 \\ \vdots \\ \mathbf{v}_n \end{array} \right] \nonumber \]

(See Exercise [ex:6_1_18].) Show that \(\{\mathbf{u}_{1}, \dots, \mathbf{u}_{n}\}\) is independent if and only if \(A\) is invertible.