midterm2-practice

# midterm2-practice - MATH 33A/2 PRACTICE MIDTERM 2 Problem...

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MATH 33A/2 PRACTICE MIDTERM 2 Problem 1. (True/False) . .. Problem 2. Let v 1 = 1 1 0 ,v 2 = 1 0 1 ,v 3 = 1 1 1 . (a) Show that B = ( v 1 ,v 2 ,v 3 ) is a basis for R 3 . (b) Let u = 1 - 1 2 . Find the coordinates of u with respect to the basis B . (c) Find the change of basis matrix from the standard basis to B . (d) Let T : R 3 R 3 be the reﬂection about the plane x + y + z = 0 . Find the matrix of T with respec to B . (e) Let T : R 3 R 3 be a linear transformation. Show that T is invertible if and only if B 0 = ( u 1 ,u 2 ,u 3 ) , where u 1 = Tv 1 , u 2 = Tv 2 and u 3 = Tv 3 , is a basis for R 3 . (f) With the same notation as in (e), assume that T is invertible. Express the change of basis matrix from the standard basis to the basis B 0 . Problem 3. (a) Prove that if V,W are subspaces of R n and V is a subspace of W , then W is a subspace of V . Prove that R n = V + V and that V V = 0 . (b) Let V R

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## This note was uploaded on 05/29/2011 for the course MATH 33a taught by Professor Lee during the Spring '08 term at UCLA.

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midterm2-practice - MATH 33A/2 PRACTICE MIDTERM 2 Problem...

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