test2-sp94 - (b) Prove that the nullspace of an m × n...

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MAS 4105 Test 2 February 8, 1994 Show all your work and provide complete details. 50 points total. 1. (8 points) Find the dimensions of the following vector spaces. (a) R 4 (b) The space of 2 × 2 symmetric matrices (c) P 3 , the space of polynomials of degree < 3 (d) { ( x 1 ,x 2 ,x 3 ,x 4 ) T R 4 | x 1 + 2 x 2 + 3 x 3 + 4 x 4 = 0 } 2. (9 points) Consider the matrices A = 1 2 1 0 0 2 5 1 1 0 3 7 2 2 - 2 4 9 3 - 1 4 B = 1 0 3 0 - 4 0 1 - 1 0 2 0 0 0 1 - 2 0 0 0 0 0 You are given that A and B are row equivalent. (a) Find the rank of A . (b) Find a basis for the nullspace of A . (c) Find a basis for the row space of A . (d) Find a basis for the column space of A . (e) Determine whether or not the rows of A are linearly independent. (f) Let the columns of A be denoted v 1 , v 2 , v 3 , v 4 , and v 5 . Which of the following sets are linearly independent? (i) { v 1 , v 2 , v 4 } (ii) { v 1 , v 3 , v 5 } (iii) { v 2 , v 4 , v 5 } 3. (8 points) (a) Define the term nullspace N ( A ) of an m × n matrix A .
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Unformatted text preview: (b) Prove that the nullspace of an m × n matrix A is a subspace of R n . 4. (8 points) (a) Define the Span of a set { v 1 ,..., v n } . (b) Let f ( x ) = x and g ( x ) = e x and consider the subspace W =Span( f,g ) of C [0 , 1]. Prove that the functions h ( x ) = x + e x and k ( x ) = x-e x span W . 1 2 5. (9 points) (a) Define the term subspace . (b) Let B be a fixed 2 × 2 matrix. Prove that the set S = { C ∈ R 2 × 2 | BC = 0 } is a subspace of R 2 × 2 . 6. (8 points) (a) Define the term linear independence . (b) Suppose v 1 , v 2 , v 3 are linearly independent vectors in a vector space V . Prove that the vectors v 1 + 2 v 2 , v 1-v 2 , v 3 are linearly independent....
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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.

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test2-sp94 - (b) Prove that the nullspace of an m × n...

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