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Unformatted text preview: MAS 4105 Test 1 (12 pts) 1. In the following let V and W be vector spaces of finite dimension and let T be a linear transformation from V into W . a. The set of all vectors S ⊂ V such that x ∈ S implies T (x) = W is called the of T . b. If G is a generating set for V containing n vectors and β is a basis for V containing m vectors, what is the relationship between n and m ? c. If dim( V ) = dim( W ) and T is onto, then rank( T ) = . d. The span of a subset of a vector space is a subspace of the vector space. True False e. A basis for a vector space V is a subset of V which the vector space and whose vectors are . f. If x, y ∈ V and c is a scalar, then since T is linear, we know T (cx + y) = + . (8 pts) 2. Let V and W be vector spaces defined over a field F and let T be a linear trans formation from V into W . Use induction to prove that if v i ∈ V and c i ∈ F for i ∈ { 1 , 2 , . . . , n } , then T n X i=1 c i v i = n X i=1 c i T (v i ) . 3. Let S be the subset of M 2 × 2 ( R ) which consists of matrices of the form a b a + b . (4 pts) a. Prove that S is a subspace of M 2 × 2 ( R ). (3 pts) b. Find a basis for S . (2 pts) c. Express the matrix 2 3 1 as a linear combination of the vectors in your basis. (1 pt) d. What is the dimension of span( S )? 4. Let S = { x 2 2x + 1 , x 4 } . (3 pts) a. Prove that S is a linearly independent subset of P 2 ( R ) . (3 pts) b. Using the standard basis of P 2 ( R ) and the techniques of Gaussian elimination, extend S to a basis of P 2 ( R ) . (8 pts) 5. Let V and W be vector spaces defined over a field F and let T and U be linear transformations from V into W . If we define ( U + T )(x) = U (x) + T (x) for x ∈ V , prove that the transformation ( U + T ) is linear. 6. The following questions pertain to the proof of Theorem 2.3 which is given on the following6....
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This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.
 Spring '09
 RUDYAK

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