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Homework 2
U = (x, y, z, w) R4 x - y - z = 0, z - w = 0 , W = (x, y, z, w) R4 y - z + w = 0, x - z = 0 , V = (x, y, z, w) R4 - x + y + w = 0 .
1. Let U, V, W R4 be the subspaces
a. Prove that V = U + W . Proving that U + W V is easier, but to show that
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homework 4, due wednesday, 5/4 in class, or in Vladimir's box by 4:00pm
1.axler chapter 1 numbers 1, 2, chapter 3 number 11, and chapter 5 number 1. 2. Let V C3 be the subspace V = cfw_(x, y, z) x + y + z = 0. Let T : C3 C3 be the operator T (x, y, z) =
Math 334 Midterm 1 Do only 3 of the four problems 2-5. Do all work and record all answers in the blue book, with your name on the front. 1. (20 points) a. If V is a vector space, define what it mean for a subset U V to be a subspace. b. For each of the fo
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homework 7 due friday, may 27 in class or by 4:00pm in vladimir's box
1. axler chapter 7 # 1,6,11,22 2. suppose V is an inner-product space (real or complex), U V is a subspace, and P rojU : V U is the projection onto U . show that, for any u U , and v
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Homework 6
1. let V = R2 . none of the the following are inner-products. for each one, give, with explanation, an axiom which is violated. a. (u, v) = u1 v1 - u2 v2 b. (u, v) = u1 v1 + u2 v1 + u2 v2 c. (u, v) = 0 d. (u, v) = u1 v2 - 5. 2. let V = C3 , w
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1. let
homework 4, due wednesday, 5/4 in class, or in Vladimir's box by 4:00pm
V = cfw_f f (x, y) is a polynomial of degree n with complex coefficients
and let T : V V be the operator (T f )(x, y) = f (y, x). a. show that every eigenvalue of T is 1 or -
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1.1
homework 1, due wednesday, 4/27 in class, or in Vladimir's box by 4:00pm
homework
V = f f (x) = a0 + a1 x + a2 x2 + a3 x3
1. Let which has basis B = cfw_1, x, x2 , x3 . also, let (T f )(x) = f (x), (Sf )(x) = f (x - 1)
a. find the matrices MB,B (S),
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homework 1, due wednesday, 4/13 in class, or in Vladimir's box by 4:00pm
1. axler chapter 1 #2, 4, 5, 10, 13
2. Let U R3 be the subspace U = cfw_(a, b, c) such that a + b + c = 0 . Find a basis of U , and prove that it is a basis using the definitions.
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