s11_mthsc853_hw03

s11_mthsc853_hw03 - map R : U V such that RS = T . Prove...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 3 Due Monday, February 7th, 2011 (1) Let T : X U be a linear map. Prove the following: (a) The image of a subspace of X is a subspace of U . (b) The inverse image of a subspace of U is a subspace of X . (2) Prove Theorem 3.3 in Lax: (a) The composite of linear mappings is also a linear mapping. (b) Composition is distributive with respect to the addition of linear maps, that is, ( R + S ) T = R T + S T and S ( T + P ) = S T + S P,, where R,S : U V and P,T : X U . (3) Suppose that X , U , and V are vector spaces, and S : X U and T : X V are linear maps. Give necessary and sufficient conditions for the existence of a well-defined linear
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Unformatted text preview: map R : U V such that RS = T . Prove all of your claims. (4) Let X be a nite-dimensional vector space over K and let { x 1 ,...,x n } be an ordered basis for X . Let U be a vector space over the same eld K but possibly with a dierent dimension, and let { u 1 ,...,u n } be an arbitrary set of vectors in U . Show that there is precisely one linear transformation T : X U such that Tx i = u i for each i = 1 ,...,n ....
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