Section 1C Subspaces; 24 problems: p 25
Sums of subspaces are analogous to unions of subsets. Similarly, direct sums of subspaces are analogous to disjoint unions of subsets. No two sub- spaces of a vector space can be disjoint, because both contain 0. So disjointness is replaced, at least in the case of two subspaces, with the requirement that the intersection equal {0}.
1. [[1C-1]] Β For each of the following subsets of π
3, determine whether it is a subspace of π
3.
(a) {(π₯1,π₯2,π₯3)βπ
3 βΆπ₯1 +2π₯2 +3π₯3 =0}
(b) {(π₯1,π₯2,π₯3)βπ
3 βΆπ₯1 +2π₯2 +3π₯3 =4}
(c) {(π₯1,π₯2,π₯3)βπ
3 βΆπ₯1π₯2π₯3 =0}
(d) {(π₯1,π₯2,π₯3) β π
3 βΆ π₯1 = 5π₯3}
2. [[1C-2 ]]Β Verify all assertions about subspaces in Example 1.35.
3. [[1C-3]] Β Show that the set of differentiable real-valued functions π on the interval (β4, 4) such that π β²(β1) = 3π(2) is a subspace of π(β4, 4).
4. [[1C-4]] Β Suppose π β $mathbb{R}$. Show that the set of continuous real-valued functions π on the interval [0, 1] such that β«1 π = π is a subspace of $mathbb{R}$[0, 1] if and only if π = 0.
5. [[1C-5]] Β Is $\mathbb{R}^2$ a subspace of the complex vector space π2?
6. [[1C-6]] Β (a) Is {(π,π,π)β$\mathbb{R}^3$ βΆπ3 =π3} a subspace of$\mathbb{R}^3$? (b) Is {(π,π,π) β π3 βΆ π3 = π3} a subspace of π3?
7. [[1C-7 ]]Β Prove or give a counterexample: If π is a nonempty subset of $\mathbb{R}^2$ such that π is closed under addition and under taking additive inverses (meaning βπ’ β π whenever π’ β π), then π is a subspace of $\mathbb{R}^2$.
8. [[1C-8]] Β Give an example of a nonempty subset π of $\mathbb{R}^2$ such that π is closed under scalar multiplication, but π is not a subspace of $\mathbb{R}^2$.
9. [[1C-9]] Β A function π βΆ $\mathbb{R}$ β $\mathbb{R}$ is called periodic if there exists a positive number π such that π(π₯) = π(π₯ + π) for all π₯ β $\mathbb{R}$ Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}$\mathbb{R}$ ? Explain.
10. [[1C-10]] Β Suppose π1 and π2 are subspaces of π. Prove that the intersection π1 β© π2 is a subspace of π.
11. [[1C-11]] Β Prove that the intersection of every collection of subspaces of π is a subspace of π.
12. [[1C-12]] Β Prove that the union of two subspaces of π is a subspace of π if and only if one of the subspaces is contained in the other.
13. [[1C-13]] Β Prove that the union of three subspaces of π is a subspace of π if and only if one of the subspaces contains the other two.
This exercise is surprisingly harder than Exercise 12, possibly because this exercise is not true if we replace π
with a field containing only two elements.
14. [[1C-14]] Β Suppose π={(π₯,βπ₯,2π₯)βπ
3 βΆπ₯βπ
} and π={(π₯,π₯,2π₯)βπ
3 βΆπ₯βπ
}.
Describe π + π using symbols, and also give a description of π + π that uses no symbols.
15. [[1C-15]] Β Suppose π is a subspace of π.What is π+π ?
16. [[1C-16]] Β Is the operation of addition on the subspaces of π commutative? In other words, if π and π are subspaces of π, is π+π = π+π?
17. [[1C-17]] Β Is the operation of addition on the subspaces of π associative? In other words, if π1, π2, π3 are subspaces of π, is (π1 +π2)+π3 =π1 +(π2 +π3)?
18. [[1C-18]] Β Does the operation of addition on the subspaces of π have an additive identity? Which subspaces have additive inverses?
19. [[1C-19]] Β Prove or give a counterexample: If π1, π2, π are subspaces of π such that π1 + π = π2 + π, then π1 = π2.
20. [[1C-20]] Β Suppose π = {(π₯,π₯,π¦,π¦) β π
4 βΆπ₯,π¦ β π
}. Find a subspace π of π
4 such thatπ
4 =πβπ.
[[1C-21]] Suppose π={(π₯,π¦,π₯+π¦,π₯βπ¦,2π₯)βπ
5 βΆπ₯,π¦βπ
}.
Find a subspace π of π
5 such that π
5 = πβπ.
[[1C-22]] Suppose π = {(π₯,π¦,π₯+π¦,π₯βπ¦,2π₯) βπ
5 βΆπ₯,π¦βπ
}.
Find three subspaces π1, π2, π3 of π
5, none of which equals {0}, such that π
5 = π β π1 β π2 β π3.
[[1C-23]] Prove or give a counterexample: If π1, π2, π are subspaces of π such that π=π1βπ and π=π2βπ, then π1 = π2.
Hint: When trying to discover whether a conjecture in linear algebra is true or false, it is often useful to start by experimenting in π
2.
[[1C-24]] A function πβΆ $\mathbb{R}$ β $\mathbb{R}$ is called even if π(βπ₯)= π(π₯)
for all π₯ β $\mathbb{R}$. A function πβΆ $\mathbb{R}$ β $\mathbb{R}$ is called odd if π(βπ₯) = βπ(π₯) for all π₯ β $\mathbb{R}$. Let πe denote the set of real-valued even functions on $\mathbb{R}$ and let πo denote the set of real-valued odd functions on $\mathbb{R}$. Show that $\mathbb{R}$\mathbb{R}$ = πe β πo.
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\begin{aligned}
1 & \text{ For each of the following subsets of } \mathbb{F}^3, \text{ determine whether it is a subspace of } \mathbb{F}^3. \\\\
& (a) \{(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 0\} \\
& (b) \{(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 4\} \\
& (c) \{(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1x_2x_3 = 0\} \\
& (d) \{(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 = 5x_3\} \\\\
2 & \text{ Verify all assertions about subspaces in Example 1.35.} \\\\
3 & \text{ Show that the set of differentiable real-valued functions } f \text{ on the interval } (-4, 4)\\& \text{ such that } f'(-1) = 3f(2) \text{ is a subspace of } \mathbb{R}(-4, 4). \\\\
4 & \text{ Suppose } b \in \mathbb{R}. \text{ Show that the set of continuous real-valued functions } f \\&\text{ on the interval } [0, 1] \\&\text{ such that } \int_0^1 f = b \text{ is a subspace of } \mathbb{R}[0, 1] \text{ if and only if } b = 0. \\\\
5 & \text{ Is } \mathbb{R}^2 \text{ a subspace of the complex vector space } \mathbb{C}^2? \\\\
6 & (a) \text{ Is } \{(a, b, c) \in \mathbb{R}^3 : a^3 = b^3\} \text{ a subspace of } \mathbb{R}^3? \\
& (b) \text{ Is } \{(a, b, c) \in \mathbb{C}^3 : a^3 = b^3\} \text{ a subspace of } \mathbb{C}^3? \\\\
7 & \text{ Prove or give a counterexample: If } U \text{ is a nonempty subset of } \mathbb{R}^2 \\&\text{ such that } U \text{ is closed under addition and under taking additive inverses, then } U \\&\text{ is a subspace of } \mathbb{R}^2. \\\\
8 & \text{ Give an example of a nonempty subset } U \text{ of } \mathbb{R}^2 \\&\text{ such that } U \text{ is closed under scalar multiplication, but } U \text{ is not a subspace of } \mathbb{R}^2. \\\\
9 & \text{ A function } f : \mathbb{R} \rightarrow \mathbb{R} \text{ is called periodic if there exists a positive number } p \\&\text{ such that } f(x) = f(x + p) \text{ for all } x \in \mathbb{R}. \\&\text{ Is the set of periodic functions from } \mathbb{R} \text{ to } \mathbb{R} \text{ a subspace of } \mathbb{R}^\mathbb{R}? \text{ Explain.} \\\\
10 & \text{ Suppose } V_1 \text{ and } V_2 \text{ are subspaces of } V.\\& \text{ Prove that the intersection } V_1 \cap V_2 \text{ is a subspace of } V.\\\\
11 & \text{ Prove that the intersection of every collection of subspaces of } V \text{ is a subspace of } V. \\\\
12 & \text{ Prove that the union of two subspaces of } V \text{ is a subspace of } V \\&\text{ if and only if one of the subspaces is contained in the other.} \\\\
13 & \text{ Prove that the union of three subspaces of } V \text{ is a subspace of } V \\&\text{ if and only if one of the subspaces contains the other two.} \\\\
14 & \text{ Suppose } U = \{(x, -x, 2x) \in \mathbb{F}^3 : x \in \mathbb{F}\} \\&\text{ and } W = \{(x, x, 2x) \in \mathbb{F}^3 : x \in \mathbb{F}\}. \\
& \text{ Describe } U + W \text{ using symbols, and also give a description of } U + W \\&\text{ that uses no symbols.} \\\\
15 & \text{ Suppose } U \text{ is a subspace of } V. \text{ What is } U + U? \\\\
16 & \text{ Is the operation of addition on the subspaces of } V \\&\text{ commutative? In other words, if } U \text{ and } W \text{ are subspaces of } V, \text{ is } U + W = W + U? \\\\
17 & \text{ Is the operation of addition on the subspaces of } V \\&\text{ associative? In other words, if } V_1, V_2, V_3 \\&\text{ are subspaces of } V, \text{ is } (V_1 + V_2) + V_3 = V_1 + (V_2 + V_3)? \\\\
18 & \text{ Does the operation of addition on the subspaces of } V \\&\text{ have an additive identity? Which subspaces have additive inverses?} \\\\
19 & \text{ Prove or give a counterexample: If } V_1, V_2, U \text{ are subspaces of } V\\& \text{ such that } V_1 + U = V_2 + U, \text{ then } V_1 = V_2. \\\\
20 & \text{ Suppose } U = \{(x, x, y, y) \in \mathbb{F}^4 : x, y \in \mathbb{F}\}. \\&\text{ Find a subspace } W \text{ of } \mathbb{F}^4 \text{ such that } \mathbb{F}^4 = U \oplus W. \\\\
21 & \text{ Suppose } U = \{(x, y, x + y, x - y, 2x) \in \mathbb{F}^5 : x, y \in \mathbb{F}\}. \\\\
& \text{ Find a subspace } W \text{ of } \mathbb{F}^5 \text{ such that } \mathbb{F}^5 = U \oplus W. \\
22 & \text{ Suppose } U = \{(x, y, x + y, x - y, 2x) \in \mathbb{F}^5 : x, y \in \mathbb{F}\}. \\
& \text{ Find three subspaces } W_1, W_2, W_3 \text{ of } \mathbb{F}^5, \text{ none of which equals } \{0\}, \\
& \text{ such that } \mathbb{F}^5 = U \oplus W_1 \oplus W_2 \oplus W_3. \\\\
23 & \text{ Prove or give a counterexample: If } V_1, V_2, U \text{ are subspaces of } V \\&\text{ such that } V = V_1 \oplus U \text{ and } V = V_2 \oplus U, \\
& \text{ then } V_1 = V_2. \\
& \text{ Hint: When trying to discover whether a conjecture in linear algebra is true or false,}\\&\text{ it is often useful to start by experimenting in } \mathbb{F}^2. \\\\
24 & \text{ A function } f : \mathbb{R} \rightarrow \mathbb{R} \text{ is called even if} \\
& f(-x) = f(x) \text{ for all } x \in \mathbb{R}.\\& \text{ A function } f : \mathbb{R} \rightarrow \mathbb{R} \text{ is called odd if} \\
& f(-x) = -f(x) \text{ for all } x \in \mathbb{R}. \\&\text{ Let } V_e \text{ denote the set of real-valued even functions on } \mathbb{R} \\&\text{ and let } V_o \text{ denote the set of real-valued odd functions on } \mathbb{R}. \\
& \text{ Show that } \mathbb{R}^\mathbb{R} = V_e \oplus V_o.
\end{aligned}
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