lect136_9_w09

# lect136_9_w09 - Friday January 23 - Lecture 9 : Nullspace,...

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Friday January 23 Lecture 9 : Nullspace, columnspace and range of a matrix. (Refers to section 3,3) Objectives: . 1. Find Null A , Col A (Im A ) of a matrix A . Similarly find Null T , Im T of a linear transformation. . 2. Define the span of a set of vectors { v 1 , v 2 ,..., v m }, span{ v 1 , v 2 ,..., v m }. Define spanning family. 3. Find a spanning family for Null A and Col A . 9.1 Recall – Recall that, for the matrix A n x m , A x = b denotes a system of linear equations. Viewed as A x = b it is a matrix equation. If b = 0 , A x = 0 is called a homogeneous system . The dimensions of the matrix A tell us that this homogeneous system must have m variables (or unknowns) and is made of n equations. Furthermore, A can also be viewed as a function called a linear transformation (function) which maps points x in R m to points in R n . The solution set of A x = b is the complete solution to the system. The complete solution to the homogeneous system A x = 0 is called the Null space of A and is denoted by Null A . 9.1.1 Example Find Null( A ) for the following matrix A . 2 1 3 1 1/2 3/2 4 2 6

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We seek the solution to A x = 0 where x = ( x 1 , x 2 , x 3 ) . For A in RRE F we have: 1 1/2 3/2 0 0 0 0 0 0 Hence the solution is x = ( x 1 , x 2 , x 3 ) = ( (− 1/2) x 2 (3/2) x 3 , x 2 , x 3 ) = ( (− 1/2) x 2 , x 2 , 0) + ( (3/2) x 3 , 0, x 3 ) = x 2 ( 1/2 , 1 , 0) + x 3 ( 3/2 , 0, 1) Thus Null( A ) is the set Null( A ) = { ( x 1 , x 2 , x 3 ) : ( x 1 , x 2 , x 3 ) = x 2 ( 1/2 , 1 , 0) + x 3 ( 3/2 , 0, 1), x 2 , x 3 in R }. We can also write: Null( A ) = { x in R 3 : x = α ( 1/2 , 1 , 0) + β ( 3/2 , 0, 1), α , β ranges over R }. 9.1.2 Remarks The nullspace of a matrix A is always a linear combination of vectors. The nullspace of a matrix
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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_9_w09 - Friday January 23 - Lecture 9 : Nullspace,...

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