CHAPTER II DIMENSION In the present chapter we investigate ...

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EXERCISE SET II. 2. Review Questions. Can I give all technical aspects in defining the dimension of a vector space? Can I make each of the following vague statements precise and prove it? “Spanning sets have more (at worse equal number of) vectors than independent set.” “A spanning set can be trimmed down to a basis.” “An independent set can be extended to a basis.” “An independent set is a basis of its span.” Can I explain the identity “rank + nullity = dimension” in detail? Do I know how to prove this identity? Drills 1. Give the dimension of each of the following vector spaces over R. (a) R 4 , (b) P5, (c) M3,4, (d) FX with X = {1, 2, 3, 4}. 2. What is the dimension of C when it is considered as a vector space over R? What about C n , also considered as a vector space over R? 3. True or False: (a) A linear homogeneous system of 100 equations with 99 unknowns (or variables) must have a nontrivial solution. (b) A linear homogeneous system of 100 equations with 200 unknowns must have a nontrivial solution. (c) 100 vectors in a subspace spanned by 50 vectors must be linearly dependent. (d) If a subspace M is spanned by 50 linearly independent vectors, then the dimension of M is at most 50. (e) If vectors v0, v1, v2, . . . , v50 are linearly independent, then the dimension of the subspace M spanned by them is 50. (f) If v1, v2 and v3 span a 2-dimensional vector space V , then one of the following three sets is a basis of V : S1 = {v2, v3}, S2 = {v1, v3}, S3 = {v1, v2}. (g) If v1, v2, v3, . . . , v99 are linearly independent vectors in a vector space V with dim V = 100, then we can pick a vector v100 in V such that v1, v2, . . . , v99, v100 form a basis of V . 24
(h) If S is a set of 123 vectors spanning a vector space V with dim V = 100, then we can delete 23 vectors from S such that the remaining vectors form a basis of V . (i) If dim V = 100 and v1, . . . , v123 span V , then v1, . . . , v100 form a basis of V . 4. Answer the following questions: (a) For two subspaces H and K of a vector space V , if the dimensions of H ∩ K, H and H + K are 2, 4 and 5 respectively, what is the dimension of K? (b) If M and N are subspaces of P10 with dim M = 7, dim N = 8 and dim(M ∩N) = 5, is M + N = P10? (c) Describe linear operators of rank zero. (d) If the kernel of a linear transformation T defined on P4 consists of polynomials of the form a + bx + (a + b + c)x 2 + cx 3 + ax 4 , where a, b, c are arbitrary scalars, what is the rank of T ? 5. Find the rank and the nullity of each of the following linear transformations. In each case, verify the identity “nullity + rank = dimension of the domain”. (a) T1 : F 2 → F 3 sending (x1, x2) to (0, x1, x2). (b) T2 : F 3 → F 3 sending (x1, x2, x3) to (0, x1, x2). (c) Sω : R 3 → R 3 given by Sω(x) = ω × x, where ω is a fixed nonzero vector in R 3 . (d) D : Pn → Pn sending p(x) ∈ Pn to its derivative p ′ (x). (e) M : P3 → P5, sending p(x) to x 2 p(x). (f) E : P2 → P2, sending p(x) to 1 2 (p(x) + p(−x)). (g) Q : P2 → P2, sending p(x) to 1 2 (p(x) − p(−x)). 1 (h) δ : M2,2 → M2,2, given by δ(X) = AX − XA, where A = 0 0 2 1 (i) Φ : M2,2 → M2,2 given by Φ(X) = BXC, where B = 1 0 1 , C = 0 0 1 . 0 6. True or false (S and T are linear operators on a finite dimensional vector space V of dimension at least 2): (a) If ST = T S and if v ∈ ker(T ), then S(v) ∈ ker(T ). (b) T and T 2 always have the same kernel. (c) If T and T 2 have the same rank, then they have the same range. (d) If T and T 2 have the same rank, then they have the same kernel. (e) The identity rank(S + T )=rank(S)+rank(T ) − rank(ST ) holds. 25 .
Page 1 and 2: CHAPTER II DIMENSION In the present
Page 3 and 4: ♠ Aside. What is supposed to be p
Page 5 and 6: The set L of all linear combination
Page 7 and 8: combination of them, that is, v can
Page 9 and 10: Applying elementary row operations
Page 11 and 12: EXERCISE SET II. 1. Review Question
Page 13 and 14: Exercises 1. Let T be a linear map
Page 15 and 16: has n + 1 elements. Now we determin
Page 17 and 18: first, the other letting j run firs
Page 19 and 20: Here, M + N stands for the subspace
Page 21 and 22: 2.5. Consider the following general
Page 23: q(x) be in ker T . Then aq(x + 1) =
Page 27 and 28: 6. Suppose that T is a linear opera
Page 29 and 30: Hence dim(M + N) + dim M ∩ N = di
Page 31: (x, y) with x = x1/x0, y = x2/x0 on

CHAPTER II DIMENSION In the present chapter we investigate ...

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