5.11.1.1E: Examples and Basic Properties Exercises

Exercises for 1

solutions

2

Let \(V\) denote the set of ordered triples \((x, y, z)\) and define addition in \(V\) as in \(\mathbb{R}^3\). For each of the following definitions of scalar multiplication, decide whether \(V\) is a vector space.

1. \(a(x, y, z) = (ax, y, az)\)
2. \(a(x, y, z) = (ax, 0, az)\)
3. \(a(x, y, z) = (0, 0, 0)\)
4. \(a(x, y, z) = (2ax, 2ay, 2az)\)

2. No; S5 fails.
3. No; S4 and S5 fail.

Are the following sets vector spaces with the indicated operations? If not, why not?

1. The set \(V\) of nonnegative real numbers; ordinary addition and scalar multiplication.
2. The set \(V\) of all polynomials of degree \(\geq 3\), together with \(0\); operations of \(\mathbf{P}\).
3. The set of all polynomials of degree \(\leq 3\); operations of \(\mathbf{P}\).
4. The set \(\{1, x, x^{2}, \dots\}\); operations of \(\mathbf{P}\).
5. The set \(V\) of all \(2 \times 2\) matrices of the form \(\left[ \begin{array}{cc} a & b \\ 0 & c \end{array} \right]\); operations of \(\mathbf{M}_{22}\).
6. The set \(V\) of \(2 \times 2\) matrices with equal column sums; operations of \(\mathbf{M}_{22}\).
7. The set \(V\) of \(2 \times 2\) matrices with zero determinant; usual matrix operations.
8. The set \(V\) of real numbers; usual operations.
9. The set \(V\) of complex numbers; usual addition and multiplication by a real number.
10. The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^2\), but using scalar multiplication \(a(x, y) = (ax, -ay)\).
11. The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^2\), but using scalar multiplication \(a(x, y) = (x, y)\) for all \(a\) in \(\mathbb{R}\).
12. The set \(V\) of all functions \(f\) : \(\mathbb{R} \to \mathbb{R}\) with pointwise addition, but scalar multiplication defined by \((af)(x) = f(ax)\).
13. The set \(V\) of all \(2 \times 2\) matrices whose entries sum to \(0\); operations of \(\mathbf{M}_{22}\).
14. The set \(V\) of all \(2 \times 2\) matrices with the addition of \(\mathbf{M}_{22}\) but scalar multiplication \(*\) defined by \(a * X = aX^{T}\).

2. No; only A1 fails.
3. No.
4. Yes.
5. Yes.
6. No.
7. No; only S3 fails.
8. No; only S4 and S5 fail.

[ex:ex6_1_3] Let \(V\) be the set of positive real numbers with vector addition being ordinary multiplication, and scalar multiplication being \(a \cdot v = v^{a}\). Show that \(V\) is a vector space.

[ex:ex6_1_4] If \(V\) is the set of ordered pairs \((x, y)\) of real numbers, show that it is a vector space with addition \((x, y) + (x_{1}, y_{1}) = (x + x_{1}, y + y_{1} + 1)\) and scalar multiplication \(a(x, y) = (ax, ay + a - 1)\). What is the zero vector in \(V\)?

The zero vector is \((0, -1)\); the negative of \((x, y)\) is \((-x, -2 - y)\).

Find \(\mathbf{x}\) and \(\mathbf{y}\) (in terms of \(\mathbf{u}\) and \(\mathbf{v}\)) such that:

\( \begin{array}[t]{rlrcr} 2\mathbf{x} & + & \mathbf{y} & = & \mathbf{u} \\ 5\mathbf{x} & + & 3\mathbf{y} & = & \mathbf{v} \\ \end{array}\) \( \begin{array}[t]{rlrcr} 3\mathbf{x} & - & 2\mathbf{y} & = & \mathbf{u} \\ 4\mathbf{x} & - & 5\mathbf{y} & = & \mathbf{v} \\ \end{array}\)

2. \(\mathbf{x} = \frac{1}{7}(5\mathbf{u} - 2\mathbf{v}), \mathbf{y} = \frac{1}{7}(4\mathbf{u} - 3\mathbf{v})\)

In each case show that the condition \(a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}\) in \(V\) implies that \(a = b = c = 0\).

1. \(V = \mathbb{R}^4\); \(\mathbf{u} = (2, 1, 0, 2)\), \(\mathbf{v} = (1, 1, -1, 0)\), \(\mathbf{w} = (0, 1, 2, 1)\)
2. \(V =\|{M}_{22}\); \(\mathbf{u} = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]\), \(\mathbf{w} = \left[ \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right]\)
3. \(V =\|{P}\); \(\mathbf{u} = x^{3} + x\), \(\mathbf{v} = x^{2} + 1\), \(\mathbf{w} = x^{3} - x^{2} + x + 1\)
4. \(V =\|{F}[0, \pi]\); \(\mathbf{u} = \sin x\), \(\mathbf{v} = \cos x\), \(\mathbf{w} = 1\)—the constant function

2. Equating entries gives \(a + c = 0\), \(b + c = 0\), \(b + c = 0\), \(a - c = 0\). The solution is \(a = b = c = 0\).
3. If \(a \sin x + b \cos y + c = 0\) in \(\mathbf{F}[0, \pi]\), then this must hold for every \(x\) in \([0, \pi]\). Taking \(x = 0, \frac{\pi}{2}\), and \(\pi\), respectively, gives \(b + c = 0\), \(a + c = 0\), \(-b + c = 0\) whence, \(a = b = c = 0\).

Simplify each of the following.

1. \(3[2(\mathbf{u} - 2\mathbf{v} - \mathbf{w}) + 3(\mathbf{w} - \mathbf{v})] - 7(\mathbf{u} - 3\mathbf{v} - \mathbf{w})\)
2. \(4(3\mathbf{u} - \mathbf{v} + \mathbf{w}) - 2[(3\mathbf{u} - 2\mathbf{v}) - 3(\mathbf{v} - \mathbf{w})] \newline + 6(\mathbf{w} - \mathbf{u} - \mathbf{v})\)

2. \(4\mathbf{w}\)

Show that \(\mathbf{x} = \mathbf{v}\) is the only solution to the equation \(\mathbf{x} + \mathbf{x} = 2\mathbf{v}\) in a vector space \(V\). Cite all axioms used.

Show that \(-\mathbf{0} = \mathbf{0}\) in any vector space. Cite all axioms used.

[ex:6_1_10] Show that the zero vector \(\mathbf{0}\) is uniquely determined by the property in axiom A4.

If \(\mathbf{z} + \mathbf{v} = \mathbf{v}\) for all \(\mathbf{v}\), then \(\mathbf{z} + \mathbf{v} = \mathbf{0} + \mathbf{v}\), so \(\mathbf{z} = \mathbf{0}\) by cancellation.

[ex:6_1_11] Given a vector \(\mathbf{v}\), show that its negative \(-\mathbf{v}\) is uniquely determined by the property in axiom A5.

[ex:6_1_12]

1. Prove (2) of Theorem [thm:017797]. [Hint: Axiom S2.]
2. Prove that \((-a)\mathbf{v} = -(a\mathbf{v})\) in Theorem [thm:017797] by first computing \((-a)\mathbf{v} + a\mathbf{v}\). Then do it using (4) of Theorem [thm:017797] and axiom S4.
3. Prove that \(a(-\mathbf{v}) = -(a\mathbf{v})\) in Theorem [thm:017797] in two ways, as in part (b).

2. \((-a)\mathbf{v} + a\mathbf{v} = (-a + a)\mathbf{v} = 0\mathbf{v} = \mathbf{0}\) by Theorem [thm:017797]. Because also \(-(a\mathbf{v}) + a\mathbf{v} = \mathbf{0}\) (by the definition of \(-(a\mathbf{v})\) in axiom A5), this means that \((-a)\mathbf{v} = -(a\mathbf{v})\) by cancellation. Alternatively, use Theorem [thm:017797](4) to give \((-a)\mathbf{v} = [(-1)a]\mathbf{v} = (-1)(a\mathbf{v}) = -(a\mathbf{v})\).

[ex:6_1_13] Let \(\mathbf{v}, \mathbf{v}_{1}, \dots, \mathbf{v}_{n}\) denote vectors in a vector space \(V\) and let \(a, a_{1}, \dots, a_{n}\) denote numbers. Use induction on \(n\) to prove each of the following.

1. \(a(\mathbf{v}_{1} + \mathbf{v}_{2} + \dots + \mathbf{v}_{n}) = a\mathbf{v}_{1} + a\mathbf{v}_{2} + \dots + a\mathbf{v}_{n}\)
2. \((a_{1} + a_{2} + \dots + a_{n})\mathbf{v} = a_{1}\mathbf{v} + a_{2}\mathbf{v} + \dots + a_{n}\mathbf{v}\)

2. The case \(n = 1\) is clear, and \(n = 2\) is axiom S3. If \(n > 2\), then \((a_{1} + a_{2} + \dots + a_{n})\mathbf{v} = [a_{1} + (a_{2} + \dots + a_{n})]\mathbf{v} = a_{1}\mathbf{v} + (a_{2} + \dots + a_{n})\mathbf{v} = a_{1}\mathbf{v} + (a_{2}\mathbf{v} + \dots + a_{n}\mathbf{v})\) using the induction hypothesis; so it holds for all \(n\).

[ex:6_1_14] Verify axioms A2—A5 and S2—S5 for the space \(\mathbf{F}[a, b]\) of functions on \([a, b]\) (Example [exa:017760]).

Prove each of the following for vectors \(\mathbf{u}\) and \(\mathbf{v}\) and scalars \(a\) and \(b\).

1. If \(a\mathbf{v} = \mathbf{0}\), then \(a = 0\) or \(\mathbf{v} = \mathbf{0}\).
2. If \(a\mathbf{v} = b\mathbf{v}\) and \(\mathbf{v} \neq \mathbf{0}\), then \(a = b\).
3. If \(a\mathbf{v} = a\mathbf{w}\) and \(a \neq 0\), then \(\mathbf{v} = \mathbf{w}\).

3. If \(a\mathbf{v} = a\mathbf{w}\), then \(\mathbf{v} = 1\mathbf{v} = (a^{-1}a)\mathbf{v} = a^{-1}(a\mathbf{v}) = a^{-1}(a\mathbf{w}) = (a^{-1}a)\mathbf{w} = 1\mathbf{w} = \mathbf{w}\).

By calculating \((1 + 1)(\mathbf{v} + \mathbf{w})\) in two ways (using axioms S2 and S3), show that axiom A2 follows from the other axioms.

[ex:ex6_1_17] Let \(V\) be a vector space, and define \(V^{n}\) to be the set of all \(n\)-tuples \((\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{n})\) of \(n\) vectors \(\mathbf{v}_{i}\), each belonging to \(V\). Define addition and scalar multiplication in \(V^{n}\) as follows:

\[\begin{gathered} (\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n) + (\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n) \\ \quad = (\mathbf{u}_1 + \mathbf{v}_1, \mathbf{u}_2 + \mathbf{v}_2, \dots, \mathbf{u}_n + \mathbf{v}_n) \\ a(\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n) = (a\mathbf{v}_1, a\mathbf{v}_2, \dots, a\mathbf{v}_n)\end{gathered} \nonumber \]

Show that \(V^{n}\) is a vector space.

[ex:6_1_18] Let \(V^{n}\) be the vector space of \(n\)-tuples from the preceding exercise, written as columns. If \(A\) is an \(m \times n\) matrix, and \(X\) is in \(V^{n}\), define \(AX\) in \(V^{m}\) by matrix multiplication. More precisely, if

\[A = \left[ a_{ij} \right] \mbox{ and } X = \left[ \begin{array}{c} \mathbf{v}_1 \\ \vdots \\ \mathbf{v}_n \end{array} \right], \mbox{ let } AX = \left[ \begin{array}{c} \mathbf{u}_1 \\ \vdots \\ \mathbf{u}_n \end{array} \right] \nonumber \]

where \(\mathbf{u}_{i} = a_{i1}\mathbf{v}_{1} + a_{i2}\mathbf{v}_{2} + \dots + a_{in}\mathbf{v}_{n}\) for each \(i\). Prove that:

1. \(B(AX) = (BA)X\)
2. \((A + A_{1})X = AX + A_{1}X\)
3. \(A(X + X_{1}) = AX + AX_{1}\)
4. \((kA)X = k(AX) = A(kX)\) if \(k\) is any number
5. \(IX = X\) if \(I\) is the \(n \times n\) identity matrix
6. Let \(E\) be an elementary matrix obtained by performing a row operation on the rows of \(I_{n}\) (see Section [sec:2_5]). Show that \(EX\) is the column resulting from performing that same row operation on the vectors (call them rows) of \(X\). [Hint: Lemma [lem:005213].]