s11_mthsc853_hw02

s11_mthsc853_hw02 - MthSc 853 (Linear Algebra) Dr. Macauley...

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Unformatted text preview: MthSc 853 (Linear Algebra) Dr. Macauley HW 2 Due Wednesday, February 2nd, 2011 (1) Let S be a set of vectors in a finite-dimensional vector space X . Show that S is a basis of X if every vector of X can be written in one and only one way as a linear combination of the vectors in S . (2) Let X be a finite-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 field K but possibly with a different 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 . (3) Let K be a finite field. The characteristic of K , denoted char K , is the smallest positive integer n for which 1 + 1 + + 1 | {z } n times = 0. (a) Prove that the characteristic of K is prime. (b) Show that K is a vector space over Z p , where p = char K ....
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