 # Linear Algebra Questions

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Problem 1. Look at the following picture, Suppose you know that in the picture v1 and v2 are orthogonal, e1 and e2 are orthogonal. (We will define what is orthogonal 2 weeks later. Now just use your geometric intuition.) 1 Now suppose there is a mirror set on span(v2). We define T to be the reflection by span(v2) (Remember span(v2) is the line that v2 stays) (0) Is this a linear map or linear transformation? (5) Find the matrix representation of T with respect to basis e1 e2 (Using previous result, do some substitution) 2 (6) Now take a look at that picture, draw −5e1, and find where is T( −5e1) in the picture. Write down the coordinate of T(−5e1) with respect to basis e1 e2 . Now use your matrix representation in previous one to algebraically calculate what is T(−5e1). Then Compare. (7) Look at the picture, tell me directly, what is the kernel of T? What is the dimension of kernel? (8) Look at the picture, tell me directly, what is the image of T?What is the dimension of image? (1) Which of the following is T v1? (A) v1 (B) − v1 (C) v2 (D) − v2 (E) 0 (2) Which of the following is T v2? (A) v1 (B) − v1 (C) v2 (D) − v2 (E) 0 (3) Find the m trix represen ation of T with respect to basis v1 v2 a (Write in the form t T v1 v2 = v1 v2 P) (4) Find the matrix P by looking at the picture, such that v1 v2 = e1 e2 P, and with your P, and do some algebraic strategy on this equation to find the matrix Q, such that e1 e2 = v1 v2 Q(Hint: what is the algebraic strategy? right multiply the inverse of so Problem 2. Suppose V, and W are two linear spaces over R, now assume e1 e2 e3 be a basis of V, and 1 2 be a basis of W. and v1 v2 v3 be bunch of vectors in V, and w1 w2 be bunch of vectors in W. and T is linear map from V to W Do the following question. I’ll show an example. . . . Suppose v1 v2 v3 = e1 e2 e3 2 1 4 5 2 7 6 1 1 ; w1 w2 = 1 2 5 4 6 5 ; Te1 = 31 + 32 Te2 = 31 + 22 Te3 = 1 + 32 1 Represent T by the basis e1 e2 e3 and 1 2 2 Determine whether v1 v2 v3 and w1 w2 are basis for domain and target respectively 3 if it is, represent linear transformation T with respect to v1 v2 v3 and w1 w2 1 Represent T by the basis e1 e2 e3 and 1 2 We know directly, T e1 e2 e3 = 1 2 3 3 1 3 2 3 2 Determine whether v1 v2 v3 and w1 w2 are basis for domain and target respectively We calculate 2 1 4 5 2 7 6 1 1 = −3 6= 0 5 4 6 5 = 1 6= 0 Thus the coordinate matrices are invertible, by the theorem, they are bases. 3 if it is, represent linear transformation T with respect to v1 v2 v3 and w1 w2 . We want to express Tv1, Tv2, and Tv3 in terms of w1 and w2 . Firstly let’s represent 1 and 2 in terms of w1 and w2 Because w1 = 51 + 62 w2 = 41 + 52 From above, we express 1, 2 in terms of w1 and w2 1 = 5w1 − 4 Now we are ready: Tv1 = T(2e1 + 5e2 + 6e3) = 2Te1 + 5Te2 + 6Te3 = 2(31 + 32) + 5(31 + 22) + 6(1 + 32) = 221 + 192 = 22(5w1 − 4w2) + 19(−6w1 + 5w2) = −w1 + 8w2 And doing the same type of calculation, we get the expression of Tv2 and Tv3, to sum up we have Tv1 = −w1 + 8w2 Tv2 = 10w1 − 10w2 Tv3 = 54w1 − 59w2 So the matrix representation of T with basis v1 v2 v3 and w1 w2 is T v1 v2 v3 = w1 w2 −1 10 54 8 −10 −59 With understanding of the method, do the exercises in the following pages. 6 (1) v1 v2 v3 = e1 e2 e3 1 1 1 1 1 ; w1 w2 = 1 2 2 1 1 1 ; Te1 = 21 + 2 Te2 = 1 Te3 = 2 e 1 e2 e3 1 Represent linear map T by the basis 2 Determine whether v1 v2 v3 and w1 w2 and12 are basis for domain and target respectively 3 if it is, represent linear transformation T with respect to v1 v2 v3 and Problem 3. How is everything going so far? Feel tons of calculation? Now we use matrix to redo the 3rd part of the previous problem, look here. v1 v2 v3 = e1 e2 e3 2 1 4 5 2 7 6 1 1 w1 w2 = 1 2 5 4 6 5 ; Te1 = 31 + 32 Te2 = 31 + 22 Te3 = 1 + 32 Now we want to represent T with respect to bases v1 v2 v3 and w1 w2 . That is easy. We know T e1 e2 e3 12 3 3 1 3 2 3 and 1 2 = w1 w2 5 4 6 5 −1 We do the following steps T v1 v2 v3 = T e1 e2 e3 2 1 4 5 2 7 6 1 1 = 1 2 3 3 1 3 2 3 2 1 4 5 2 7 6 1 1 = w1 w2 5 4 6 5−1 3 3 1 3 2 3 2 1 4 5 2 7 6 1 1 = w1 w2 −1 10 54 8 −10 −59 So that means the matrix representation of T with respect to v1 v2 v3 and w1 w2 is T v1 v2 v3 = w1 w2 −1 10 54 8 −10 −59 Do the problem on the next page again usi (1) v1 v2 v3 = e1 e2 e3 1 1 5 2 1 3 1 w1 w2 = 12 1 1 −1 1 ; Te1 = 41 + 22 Te2 = 21 + 22 Te3 = 22 represent linear transformation T with respect to v1 v2 v3 and w1 w2 Problem 4. Let V = M3×1(R) denote the space of 3 × 1 matrices over R, W = M2×1(R) denote the space of 2 × 1 matrices over R, T is a linear map T : V −→ W And suppose we know: T 1 2 5 1 2 ; T 1 3 4 5 ; T −2 1 2 1 0 ; We want to realize T as a matrix multiplication. (1) Show that { 1 2 5 4 1 3 −2 1 2 } is a basis for V (Write down the coordinate matrix of this bunch of vectors with respect to natural basis. And try to show that coordinate matrix is invertible.) (2) Now with basis 1 2 5 4 1 3 −2 1 2 in the domain, and natural basis in the target. Write down the matrix representation of linear transformation T. 13 e (3) R lize linear tr a ansformation T as a matrix multiplication. (Hint, you have the form like T e1 e2 e3 = 1 2 P, now because ei and i are column matrices, thus all the thing except T could view as block matrices. Now treat e1 e2 e3 as a matrix, and move it to the right. You get an expression of T equal to a matrix product. Then this realizes T as a matrix ) (4) With your realization, calculate T 2 3 1 Problem 5. Suppose V = Px 2 = {ax2 + bx + c|a, b, c ∈ R}, W = Pt 2 = {at2 + bt + c|a, b, c ∈ R}. Plug in t = x + 1 for polynomial in V gives an polynomial in W. Thus defines a linear map T. For example T(3x 2 + 2x + 3) = 3(t + 1)2 + 2(t + 1) + 3 = 3t 2 + 8t + 8 (1) Is this a linear map or linear transformation? (Hint: t,x is different letter) (2) With basis x 2 x 1 in domain, and basis t 2 t 1 in target, write down the matrix representation of linear map T.(present in the form T x 2 x 1 = t 2 t 1 P) (3) Now using your matrix representation, find the coordinate of T(x 2 + 5x + 1) with respect to basis t 2 t 1 (In this problem, only matrix multiplication is allowed) (4) Find the kernel of this linear map. (5) Show that T is an isomorphism Problem 6. We want to find out all 2 × 2 matrices that can commute with 1 2 . Suppose V = M2×2(R) the linear space of 2 × 2 matrices. Define a linear transformation: T : V −→ V T(M) = M 1 2 − 1 2  M (1) With respect to the basis consisting of  1 1 , 1 −1 , 1 , 1 Find the matrix representation of this linear transformation. (2) The matrices commuting with 1 forms a subspace in V. Can you find a basis of this 2 subspace? What is the dimension of the subspace? (Think how does this question related to kernel of T) (3) The matrices P, that can be represented as P = M 1 − 1 M for some M, 2 2 forms a subspace in V. Can you find a basis of the subspace? What is the dimension of the subspace? (Think how does this question related to image Problem 7. There are 3 brothers. Namely, A,B,C. They have some money originally. Suppose there is neither income and costs. And they like to share their money. At the end of every month, each of them will split his money into two equal part and give it to two others. For example. Suppose A has 300 dollars at first month. At second month, A with give 150,150 to B,C respectively. and B,C also do the same thing. (0) Suppose this month, A has \$8192, B has \$4096, C has \$8192. What amount will they have after 1 year? (Please say: Give up) (1) OK, don’t worry, suppose T is the linear transformation that transform the account status this month to the status next month. With basis e1 = A dollar that belongs to A e2 = A dollar that belongs to B e3 = A dollar that belongs to C Write down the matrix representation of T with respect to e1 e2 e3 (Hint, Te1 = 1 2 e2 + 1 2 e3, A dollar belongs to A this month would become 2 quarters belongs to B and 2 quarters belongs to C next month.) (2) Now suppose 1 = e1 + e2 + e3 2 = e1 − e2 3 = e2 − e3 Show that 1 2 3 is a basis. And calculate T(1), T(2), T(3), represent your result in terms of 1, 2, 3 20 (3) Calculate T 12(1), T 12(2), T 12(3), represent T 12 with respect to basis 1 2 3 (4) Now with the above calculation, represent T 12 with respect to basis e1 e2 e3 (5) Suppose this month, A has \$8192, B has \$4096, C has \$8192. What amount will they have after 1 y

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