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

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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 0 2 2 4 0 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 . We can choose the variables x 3 , x 4 , x 5 freely and x 1 , x 2 are determined by our equations.
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