m235notes9 - Notes 9 Bases of Kernels Images Lecture Oct 6...

Info iconThis preview shows pages 1–2. 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: Notes 9: Bases of Kernels, Images Lecture Oct 6, 2009 Basis of the Kernel Definition 1. As subset S of R n that is closed under addition and scalar multiplication is said to be a subspace of R n . More concretely S is a subspace provided the two conditions below are satisfied. • If s 1 ,s 2 ∈ S , then s 1 + s 2 ∈ S. • If s ∈ S,λ ∈ R , then λs ∈ S. Example 1 . Let F : R m-→ R n be a linear transformation. Then ker ( F ) and the image of F ( which we denote by im ( F ) are both subspaces. The kernel of F is a subspace of R m , the domain, and im ( F ) is a subspace of R n , the target or range of F. Example 2 . Let T the subset of R 3 consisting of all the points of R 3 whose coordinates are all integers. Then T is closed under addition, but T is not closed under scalar multi- plication. Hence T is not a subspace. Definition 2. Let S be a subset of R n . The span of S is the set of all linear combinations of elements of S. Given a linear map we show how to find a small set S of vectors with the property that they span the kernel of our linear map. We do this in an example. Let A be the matrix 1- 1 0- 1 1 2 2 2 4 1 1 2 1 , so A gives a map from R 5 to R 3 . Upon row reduction we get 1 0 1 1 1 0 1 1 2 1 0 0 0 0 0 . Thus the kernel of A is the set of all solutions to the equations x 1 + x 3 + x 4 + x 5 = 0 x 2 + x 3 + 2 x 4 + x 5 = 0 ....
View Full Document

This note was uploaded on 11/22/2011 for the course MATH 235 taught by Professor Markman during the Fall '08 term at UMass (Amherst).

Page1 / 4

m235notes9 - Notes 9 Bases of Kernels Images Lecture Oct 6...

This preview shows document pages 1 - 2. Sign up to view the full document.

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