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Problem 1. Now, I choose a point O on your paper. Then your paper (if extend to infinity) become a linear space, denote as V (1) Suppose O was chosen in the following picture. Now v, w ∈ V , draw the vector that corresponds to this two element in the graph. (2) Suppose our linear space is a right linear space and if we have u = v + 2w, correct the historical mistake notation.(Hint: put the scalars on the right please), and then write u in the form v w ∗ ∗ , after that, draw u in the picture: Problem 2. Now suppose I choose O to be origin in the following picture, v1, v2, v3 as shown (1) Draw v1 v2 v3 −1 2 3 (2) Does v1, v2, v3 in the picture linearly independent? If not, find a non-zero 3 × 1 matrix ∗ ∗ ∗ over R such that v1 v2 v3 ∗ ∗ = 0, is this kind of matrix unique? If not, write couple more.(the equation v1 v2 v3 ∗ ∗ ∗= 0 is called a linear relation of v1 v2 v3 o (3) Write 3 more distinct column matrix ∗ ∗ ∗such that v1 v2 v3 ∗ ∗ ∗ v1 v2 v3 −1 2 3 Problem 3. Suppose e1 e2 is a basis for a linear space V over R. if u = e1 + 2e2 v = e1 − e2 (1) Write u v as e1 e2 multiplied by some matrix. (2) Is u v linearly independent? prove it. (3) Can u v spans the whole space? prove it. (4) Is u v a basis? prove it. (5) What is the coordinate of 4e1 − e2 with respect to the basis u Problem 4. Suppose V = M2×2(R), addition defined by matrix addition, multiplication defined by usual scalar multiplication. Thus V become a linear space. Determine whether the following subsets is linear subspace. Explain. (1) { a b c d | a b c d 1 2 = 0 0 } (2) { a b c d | a b c d 1 2 = 1 0 } (3) { a b c d | a b c d is invertible} 4 (4) { a b c d |rank a b c d = 0} (5) { a b c d | a b c d2 1 1 2 = 2 1 1 2 a b c d } Problem 5. Over R, Please choose an 2×2 matrix A that you like most, and with your A, complete the following exercises. (0) Write your A over here. 5 (1) Calculate A2 (2) Is I, A, A2 linearly dependent? If it is, find a non-trivial linear relation. If not, check your calculation in (1) and do this exercise again.(I is unit matrix. non-trivial linear relation means a not-all-zero scalar linear combination to produce zero vector) Problem 6. Let V be a linear space over F, suppose (e1, e2, · · · , en) and (1, 2, · · · , m) are both basis for V . (1) Does there exist unique matrix P such that (1, 2, · · · , m) = (e1, e2, · · · , en)P? Why? What is the size of P? (2) Does there exist unique matrix Q such that (e1, e2, · · · , en) = (1, 2, · · · , m)Q? Why? What is the size of Q? 6 (3) Prove that P Q = In, and QP = Im(Hint: Using equation in (2) to substitute something in (1), and using equation in (1) to substitute something in (2), you would find something. And by left cancellation rule of independent vectors, you get what you want)

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Linear Algebra Questions
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