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linalgmt1sol - Math 334 Midterm 1 Do only 3 of the four...

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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 following subsets of R 2 , say which of the axioms of a subspace are satisfied. i. V = ( x, y ) x = 2 R 2 none of them ii. V = ( x, y ) xy = 0 R 2 all except closed under addition 2. (30 points) a. Define what it means for a subset B V to be linearly independent , to span V, and to be a basis . b. Let V = P 3 , let T : V V be the operator ( Tf )( x ) = f ( x + 1) , and consider the basis B = { 1 , x, x 2 } ⊂ V . Find the matrix M B , B ( T ) of T in the basis B . 1 1 1 1 2 1 3. (30 points) a. If U, W V are subspaces, define the sum U + W . b. Suppose U, W are subspaces of R 5 of dimensions 2 and 4. Prove that U W contains a vector other than 0. by a theorem in the book, dim( U + W ) = dim( U )+dim( W ) - dim( U W ) = 6 - dim( U W ). but dim( U + W ) 5 since it’s a subspace of R 5 , so dim( U W ) 6 = 0, meaning it’s not the zero vector space.
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