09-29-10

# 09-29-10 - 1-1 5-2 5-6 5. Theorem 11 from section 4.5 of...

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Math 54 Discussion Section Problems Mike Hartglass September 29, 2010 You should work on the following problems in groups of 3 or 4. Try to get through as many as you can, but you aren’t expected to ﬁnish everything. In fact, the answers are largely unimportant; making sure everyone in your group knows how to solve all the problems is what really matters. 1. Suppose you have some vector space V, a basis B = { b 1 , b 2 , b 3 } and vector x with [ x ] B = 8 - 9 6 . Now suppose that someone gives you a new set of vectors C = { c 1 , c 2 , c 3 } and tells you that [ c 1 ] B = 1 - 1 3 , [ c 2 ] B = - 3 4 9 , [ c 3 ] B = 2 - 2 4 . (a) Is C a basis for V? (b) Find [ x ] C (c) Find the change of coordinate matrices from (i) C to B and (ii) B to C 2. Let A be some ﬁxed 3 × 4 matrix. Prove that the set { B M 4 × 2 : AB = 0 } is a subspace of M 4 × 2 . 3. Show that if A is a 30 × 36 matrix and dim Nul A = 6, then A x = b has a solution for every b R 30 4. Find a basis for the row space, the column space, and the null space of A = 1 - 4 9 - 7 - 1 2 - 4 1 5 - 6 10 7 , which is row equivalent to
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Unformatted text preview: 1-1 5-2 5-6 5. Theorem 11 from section 4.5 of the book tells us that given any linearly independent vectors { v 1 , v 2 ,... v p } in R n , we can extend it to a basis for R n by adding n-p vectors to it. However, it doesn’t really give us an idea of how to actually do this. Here’s one possible way: create the matrix A = [ v 1 v 2 ··· v p e 1 ··· e n ] ( e i are the columns of the identity matrix) and then use methods we already know to ﬁnd a basis for Col A. (a) How do you know the basis you come up with will be a basis for R n ? (b) How do you know it will include the vectors v 1 , v 2 ,... v p ? (c) Use this method to extend the following set of vectors to be a basis for R 4 : v 1 = 1 2 1 , v 2 = 3 8-2 3...
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## This note was uploaded on 11/19/2010 for the course LECTURE 1 taught by Professor Yildiz during the Fall '10 term at Berkeley.

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