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exam7 - M340L Third Midterm Exam April 7 2003 1 The...

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M340L Third Midterm Exam, April 7, 2003 1. The following two 4 × 4 matrices are row-equivalent: A = 1 1 2 1 1 3 8 0 2 4 10 1 3 5 12 0 ; B = 1 0 - 1 0 0 1 3 0 0 0 0 1 0 0 0 0 a) What is the rank of A ? b) Find a basis for the column space of A . c) Find a basis for the null space of A . d) Find a basis for the row space of A . 2. Let V =Span { 1 1 1 1 , 1 2 3 4 , 1 3 5 8 , 4 3 2 1 } ⊂ R 4 . a) What is the dimension of V ? b) Find a basis for V . c) Let W be the subspace of P 3 [ t ] spanned by the polynomials 1+ t + t 2 + t 3 , 1 + 2 t + 3 t 2 + 4 t 3 , 1 + 3 t + 5 t 2 + 8 t 3 and 4 + 3 t + 2 t 2 + t 3 . What is the dimension of W ? Find a basis for W . 3. Consider the following basis for R 3 : B = 1 2 2 , 3 7 7 , 6 12 13 a) Find the change-of-basis matrix P EB that converts from coordinates in the B basis to coordinates in the standard ( E ) basis.
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