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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 ....
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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).
 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Addition, Multiplication, Scalar

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