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# Linear Algebra Done Right

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W x; y 2 Fg; U2 D f.0; 0; z/ 2 F3 W z 2 Fg; U3 D f.0; y; y/ 2 F3 W y 2 Fg: Show that U1 C U2 C U3 is not a direct sum. Solution Clearly F3 D U1 C U2 C U3, because every vector .x; y; z/ 2 F3 can be written as .x; y; z/ D .x; y; 0/C .0; 0; z/C .0; 0; 0/; where the ﬁrst vector on the right side is in U1, the second vector is in U2, and the third vector is in U3. However, F3 does not equal the direct sum of U1; U2; U3, because the vector .0; 0; 0/ can be written in two different ways as a sum u1 C u2 C u3, with each uj in Uj . Speciﬁcally, we have .0; 0; 0/ D .0; 1; 0/C .0; 0; 1/C .0;�1;�1/ and, of course, .0; 0; 0/ D .0; 0; 0/C .0; 0; 0/C .0; 0; 0/; where the ﬁrst vector on the right side of each equation above is in U1, the second vector is in U2, and the third vector is in U3. The symbol ˚, which is a plus sign inside a circle, serves as a re- minder that we are dealing with a special type of sum of subspaces— each element in the direct sum can be represented only one way as a sum of elements from the speciﬁed subspaces. The deﬁnition of direct sum requires that every vector in the sum have a unique representation as an appropriate sum. The next result shows that when deciding whether a sum of subspaces is a direct sum, we need only consider whether 0 can be uniquely written as an appropriate sum. � 7� CHAPTER 1 Vector Spaces SECTION 1.C Subspaces 23 1.44 Condition for a direct sum Suppose U1; : : : ; Um are subspaces of V. Then U1 C � � � C Um is a direct sum if and only if the only way to write 0 as a sum u1 C � � � C um, where each uj is in Uj , is by taking each uj equal to 0. Proof First suppose U1 C � � � C Um is a direct sum. Then the deﬁnition of direct sum implies that the only way to write 0 as a sum u1 C� � �Cum, where each uj is in Uj , is by taking each uj equal to 0. Now suppose that the only way to write 0 as a sum u1 C � � � C um, where each uj is in Uj , is by taking each uj equal to 0. To show that U1 C� � �CUm is a direct sum, let v 2 U1 C � � � C Um. We can write v D u1 C � � � C um for some u1 2 U1; : : : ; um 2 Um. To show that this representation is unique, suppose we also have v D v1 C � � � C vm; where v1 2 U1; : : : ; vm 2 Um. Subtracting these two equations, we have 0 D .u1 � v1/C � � � C .um � vm/: Because u1 � v1 2 U1; : : : ; um � vm 2 Um, the equation above implies that each uj � vj equals 0. Thus u1 D v1; : : : ; um D vm, as desired. The next result gives a simple condition for testing which pairs of sub- spaces give a direct sum. 1.45 Direct sum of two subspaces Suppose U and W are subspaces of V. Then U CW is a direct sum if and only if U \W D f0g. Proof First suppose that U C W is a direct sum. If v 2 U \ W, then 0 D v C .�v/, where v 2 U and �v 2 W. By the unique representation of 0 as the sum of a vector in U and a vector in W, we have v D 0. Thus U \W D f0g, completing the proof in one direction. To prove the other direction, now suppose U \W D f0g. To prove that U CW is a direct sum, suppose u 2 U, w 2 W, and 0 D uC w: To complete the proof, we need only show that u D w D 0 (by 1.44). The equation above implies that u D �w 2 W. Thus u 2 U \W. Hence u D 0, which by the equation above implies that w D 0, completing the proof. 24 CHAPTER 1 Vector Spaces Sums of subspaces are analogous to unions of subsets. Similarly, di- rect sums of subspaces are analo- gous to disjoint unions of subsets. No two subspaces of a vector space can be disjoint, because both con- tain 0. So disjointness is replaced, at least in the case of two sub- spaces, with the requirement that the intersection equals f0g. The result above deals only with the case of two subspaces. When ask- ing about a possible direct sum with more than two subspaces, it is not enough to test that each pair of the subspaces intersect only at 0. To see this, consider Example 1.43. In that nonexample of a direct sum, we have U1 \U2 D U1 \U3 D U2 \U3 D f0g. EXERCISES 1.C 1 For each of the following subsets of F3, determine whether it is a sub- space of F3: (a) f.x1; x2; x3/ 2 F3 W x1 C 2x2 C 3x3 D 0g; (b) f.x1; x2; x3/ 2 F3 W x1 C 2x2 C 3x3 D 4g; (c) f.x1; x2; x3/ 2 F3 W x1x2x3 D 0g; (d) f.x1; x2; x3/ 2 F3 W x1 D 5x3g. 2 Verify all the assertions in Example 1.35. 3 Show that the set of differentiable real-valued functions f on the interval .�4; 4/ such that f 0.�1/ D 3f .2/ is a subspace of R.�4;4/. 4 Suppose b 2 R. Show that the set of continuous real-valued functions f on the interval Œ0; 1� such that R 1 0 f D b is a subspace of RŒ0;1� if and only if b D 0. 5 Is R2 a subspace of the complex vector space C2? 6 (a) Is f.a; b; c/ 2 R3 W a3 D b3g a subspace of R3? (b) Is f.a; b; c/ 2 C3 W a3 D b3g a subspace of C3? 7 Give an example of a nonempty subset U of R2 such that U is closed under addition and under taking additive inverses (meaning �u 2 U whenever u 2 U ), but U is not a subspace of R2. 8 Give an example of a nonempty subset U of R2 such that U is closed under scalar multiplication, but U is not a subspace of R2. SECTION 1.C Subspaces 25 9 A function f W R ! R is called periodic if there exists a positive number p such that f .x/ D f .x C p/ for all x 2 R. Is the set of periodic functions from R to R a subspace of RR? Explain. 10 Suppose U1 and U2 are subspaces of V. Prove that the intersection U1 \ U2 is a subspace of V. 11 Prove that the intersection of every collection of subspaces of V is a subspace of V. 12 Prove that the union of two subspaces of V is a subspace of V if and only if one of the subspaces is contained in the other. 13 Prove that the union of three subspaces of V is a subspace of V if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because this exercise is not true if we replace F with a ﬁeld containing only two elements.] 14 Verify the assertion in Example 1.38. 15 Suppose U is a subspace of V. What is U C U ? 16 Is the operation of addition on the subspaces of V commutative? In other words, if U and W are subspaces of V, is U CW D W C U ? 17 Is the operation of addition on the subspaces of V associative? In other words, if U1; U2; U3 are subspaces of V, is .U1 C U2/C U3 D U1 C .U2 C U3/‹ 18 Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses? 19 Prove or give a counterexample: if U1; U2; W are subspaces of V such that U1 CW D U2 CW; then U1 D U2. 20 Suppose U D f.x; x; y; y/ 2 F4 W x; y 2 Fg: Find a subspace W of F4 such that F4 D U ˚W. 26 CHAPTER 1 Vector Spaces 21 Suppose U D f.x; y; x C y; x � y; 2x/ 2 F5 W x; y 2 Fg: Find a subspace W of F5 such that F5 D U ˚W. 22 Suppose U D f.x; y; x C y; x � y; 2x/ 2 F5 W x; y 2 Fg: Find three subspaces W1; W2; W3 of F5, none of which equals f0g, such that F5 D U ˚W1 ˚W2 ˚W3. 23 Prove or give a counterexample: if U1; U2; W are subspaces of V such that V D U1 ˚W and V D U2 ˚W; then U1 D U2. 24 A function f W R ! R is called even if f .�x/ D f .x/ for all x 2 R. A function f W R ! R is called odd if f .�x/ D �f .x/ for all x 2 R. Let Ue denote the set of real-valued even functions on R and let Uo denote the set of real-valued odd functions on R. Show that RR D Ue ˚ Uo. CHAPTER 2 American mathematician Paul Halmos (1916–2006), who in 1942 published the ﬁrst modern linear algebra book. The title of Halmos’s book was the same as the title of this chapter. Finite-Dimensional Vector Spaces Let’s review our standing assumptions: 2.1 Notation F, V � F denotes R or C. � V denotes a vector space over F. In the last chapter we learned about vector spaces. Linear algebra focuses not on arbitrary vector spaces, but on ﬁnite-dimensional vector spaces, which we introduce in this chapter.