practice-Math33A-mt1

# practice-Math33A-mt1 - 1 1 1 Find a basis for S ⊥ 3 The...

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Name: Student ID: Section: 2 Prof. Alan J. Laub Apr. 25, 2008 Math 33A – MIDTERM EXAMINATION I Spring 2008 Instructions: (a) The exam is closed-book (except for one page of notes) and will last 50 minutes. (b) Notation will conform as closely as possible to the standard notation used in the lectures. (c) Do all 4 problems. Each problem is worth 25 points. Some partial credit may be assigned if warranted. (d) Label clearly the problem number and the material you wish to be graded. 1. You are given the matrix A = 1 2 0 4 1 0 1 0 2 1 0 0 0 0 2 IR 3 × 5 . (a) Put A in row-reduced echelon form (rref). (b) Find a basis for Im( A ) from rref( A ). (c) Find a basis for Ker( A ) from rref( A ). (d) What is rank( A )? What is rank( A T )?

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2. Let S = Sp 1 1 1
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Unformatted text preview: 1 1 1 . Find a basis for S ⊥ . 3. The vector •-4 7 ‚ has components 5 and-3 with respect to the basis ‰• 1 2 ‚ , • 3 1 ‚± . What are the components of the vector •-4 7 ‚ with respect to the basis ‰• 1 ‚ , • 1-1 ‚± of IR 2 ? 4. (a) Suppose A ∈ IR 19 × 48 8 , i.e., A is the coeﬃcient matrix of a system of 19 equations in 48 unknowns but only 8 of the equations are independent. How many linearly independent solutions can be found to the homogeneous linear system Ax = 0? (b) For the same A as in (a), how many linear independent solutions can be found to the homogeneous linear system A T y = 0?...
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