lect136_13_w09

lect136_13_w09 - Monday February 2 - Lecture 13: Examples...

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Monday February 2 Lecture 13: Examples on Row(A), Col(A) and Null(A) (Refers to section 3.4 Expectations: 1. Find a spanning family of a matrix A of size equal to the rank A . 2. Determine whether a vector belongs to the span of a finite family of vectors of R n . 3. Define W for a subspace W of R n . 4. Show that (Row( A )) = Null( A ). 13.1 Proposition – Let { v 1 , v 2 , …, v m } be a set of vectors in R n . Let A be a matrix whose column vectors are v 1 , v 2 , …, v m . Then a vector u belongs to the span{ v 1 , v 2 , …, v m } if and only if A x = u has a solution (equivalently, the system A x = u is consistent) . Proof: The system A x = u has a solution there exists a = ( a 1 , a 1 , …, a m ) such that A a = u a 1 v 1 + a 2 v 2 + …+ a m v m = u 13.2 Example Does the vector (1, 0, 0) belong to U = span{(2, 1 4), (1, ½, 2), (3, 3/2 6)}? Answer: We are asking if the system 2 1 3 | 1 1 1/2 3/2 | 0 4 2 6 | 0 has a solution. (Make sure you understand why we may interpret this question this way.) We row reduce to RREF and obtain
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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lect136_13_w09 - Monday February 2 - Lecture 13: Examples...

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