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MATH136-Assignment2 (2)

# MATH136-Assignment2 (2) - c u = then u = or c = 0 4 Let H...

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Math 136 Assignment #2 Winter 2008 Topics: Vector spaces and subspaces Due Date: January 23rd. Note 1: Below we always assume that all vector spaces are over R , i.e., the scalars are chosen from reals (unless otherwise specified). Below I n denotes the n × n identity matrix. Note 2: Assume vector addition and scalar multiplication follows the standard definition. 1. Say if the following is true or false. Please justify your answers. (a) Let A 1 = x y : x 0 , y 0 and A 2 = x y : x < 0 , y > 0 . Is A 1 A 2 ( A 1 union A 2 ) a subspace of R 2 ? (b) Is the set of all polynomials of degree n ( n a positive integer) in variable x , with even (integer) coefficients (including 0) a subspace of the vector space of all polynomials? 2 (a) Show that the set of all m × n matrices with complex entries denoted M m × n is a vector space. Which is the zero vector of this space? (b) Show that H = a - b b d : a, b, d, C is a subspace of M 2 × 2 . 3. Let V be a vector space. Let u V . Prove the following (using axioms of vector space): [ Hint: See exercises 25, 26, 27, 28 in Section 4.1. ] Let c be a scalar. If

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Unformatted text preview: c u = then u = or c = 0. 4. Let H be a non-empty subset of vector space V with the following prop-erties: (a) If u , v ∈ H , then 4 u – 5 v ∈ H . (b) If u ∈ H , then c u ∈ H for all scalars c ∈ R . Is H a subspace of V ? 5. Let H,K be subspaces of a vector space V . (a) Is H ∩ K ( H intersection K ) a subspace of V ? (b) Is H ∪ K ( H union K ) a subspace of V ? (c) Is A = { w =-u + 2 v : u ∈ H, v ∈ K } a subspace of V ? 1 6 (a) Let V be a vector space. Let v , u ∈ V . Show that span { v , u } is a subspace of V . (b) Let H = s + 4 t s-t 2 s 3 t : s,t, ∈ R . Is H a subspace of R 4 ? (c) Let V 1 = span 1 2 3 , -1 4 and V 2 = span 8-2 15 . Is V 2 a subspace of V 1 ? 2...
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MATH136-Assignment2 (2) - c u = then u = or c = 0 4 Let H...

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