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**Unformatted text preview: **Notes 10: Bases of Kernels, Images: Geometry 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, a subset S of R n is a subspace of R n provided the two conditions below are satisfied. If s 1 ,s 2 S , then s 1 + s 2 S. If s S, R , then s S. Example 1 . Let F be a linear map F : R m- R n . Then ker ( F ) is a subspace. Proof. Let x,y ker ( F ). We have F ( x + y ) = F ( x ) + F ( y ) since F is linear. Since F ( x ) = F ( y ) = 0, we conclude that F ( x + y ) = 0 also. This says that x + y ker ( F ) . Let x ker ( F ) , R . Since F is linear, we have F ( x ) = F ( x ) = 0 = 0. This says that x ker ( F ). We conclude that ker ( F ) is a subspace. Example 2 . We also have that im ( F ) is a subspace. Example 3 . We find all the subspaces of R . The set consisting of the element 0 is a subspace. Let S be a subspace with a non-zero element, say a . Let x be an arbitrary element of R . Since a 6 = 0 ,x is a scalar multiple of a . Thus the subspace is all of R ....

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