math110s-hw1 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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Unformatted text preview: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 1 In all your proofs, state clearly which field axioms or vector space axioms you have used to get from one step to the next. You may wish to underline the vectors (ie. elements of the vector space) to distinguish them from the scalars (ie. elements of the field) — e.g. 0 denotes the additive identity of the field while 0 denotes the additive identity of the vector space. 1. Prove that the following are vector spaces over R : (a) polynomials of degree not more than d , P d = { a + a 1 x + ··· + a d x d | a i ∈ R for all i } , (b) m-by- n matrices R m × n = { [ a ij ] m,n i,j =1 | a ij ∈ R for all i,j } . The addition and scalar multiplication operations for polynomials and matrices are as defined in the lectures. 2. Let V be a vector space over R with addition and scalar multiplication denoted by + and · respectively. Let W = V × V = { ( v 1 , v 2 ) | v 1 , v 2 ∈ V } . Prove that W is a vector space over C with addition defined by...
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This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.

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math110s-hw1 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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