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2006 Fall term test

# 2006 Fall term test - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences Term Test MATB24H Linear Algebra II Examiner: S. Tryphonas Date: November 4, 2006 Duration: 110 minutes 1. [10 points] Let V be a vector space over the field F . Let z be the zero vector of V and 0 be the zero element of F . (a) Using only the conditions listed in the definition of a vector space, prove that for each v V and a F : i. ( - a ) v = - ( a v ) . ii. a ( - v ) = - a ( v ) . (b) Determine whether the set V of all 2 x 2 matrices with the standard matrix addition and scalar multiplication defined by r A = r · A T for all A V and r R is a real vector space. 2. [10 points] (a) Give the definition of a subspace W of a real vector space V . (b) Let T : V -→ V be a linear transformation from the vector space V into the vector space V . If W is a subspace of V , prove that T [ W ], the image of W under T is a subspace of V 3. [10 points] Let V, V be vector spaces over R . Let T : V -→ V be a linear transformation. Suppose v 1 , v 2 , · · · , v k V .

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2006 Fall term test - University of Toronto at Scarborough...

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