extranotes Kernel and image

Extranotes Kernel - Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n × m matrix A = a 11 a 12 ·· a 1 m a 21 a 22 ·· a

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n × m matrix A = a 11 a 12 ··· a 1 m a 21 a 22 ··· a 2 m . . . . . . . . . a n 1 a n 2 ··· a nm and think of it as a function A : R m-→ R n . The kernel of A is defined as ker A = set of all x in R m such that A x = . Note that ker A lives in R m . The image of A is im A = set of all vectors in R n which are A x for some x ∈ R m . Note that im A lives in R n . Many calculations in linear algebra boil down to the computation of kernels and images of matrices. Here are some different ways of thinking about ker A and im A . In terms of equations, ker A is the set of solution vectors x = ( x 1 , . . . , x m ) in R m of the n equations a 11 x 1 + a 12 x 2 + ··· + a 1 m x m = 0 a 21 x 1 + a 22 x 2 + ··· + a 2 m x m = 0 . . . a n 1 x 1 + a n 2 x 2 + ··· + a nm x m = 0 , (19a) and im A consists of those vectors y = ( y 1 , . . . y n ) in R n for which the system a 11 x 1 + a 12 x 2 + ··· + a 1 m x m = y 1 a 21 x 1 + a 22 x 2 + ··· + a 2 m x m = y 2 . . . a n 1 x 1 + a n 2 x 2 + ··· + a nm x m = y n , (19b) 1 2 has a solution x = ( x 1 , . . . , x m ). A single equation a i 1 x 1 + a i 2 x 2 + ··· + a im x m = y i is called a hyperplane in R m . (So a line is a hyperplane in R 2 , and a plane is a hyperplane in R 3 .) Geometrically, ker A is the intersection of hyperplanes (19a), and im A is the set of vectors ( y 1 , . . . , y n ) ∈ R n for which the hyperplanes (19b) intersect in at least one point. If x and x are two solutions of (19b) for the same y , then A ( x- x ) = A x- A x = y- y = , so x- x belongs to the kernel of A . If ker A = 0 (i.e., consists just of the zero vector) then there can be at most one solution. In general, the bigger the kernel, the more solutions there are to a given equation that has at least one solution....
View Full Document

This note was uploaded on 11/22/2010 for the course MATH math 270 taught by Professor Xu during the Winter '07 term at McGill.

Page1 / 5

Extranotes Kernel - Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n × m matrix A = a 11 a 12 ·· a 1 m a 21 a 22 ·· a

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online