quiz_04_laqf_07_solution - V to W , and let T be a subspace...

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Linear Algebra, Fall 2007 Quiz 4 Hints 1. Let A be a 6 × 5 matrix. If a 1 , a 2 , a 4 are linearly independent and a 3 = a 1 + a 2 , a 5 = a 1 - a 2 + 2 a 4 , find the reduced row echelon form of A and a basis of the column space of A . The proof of Theorem 3.6.5 (and Example 3) explains how to find a basis for the row space of A and the null space of A . The paragraph before Theorem 3.6.6, the proof of Theorem 3.6.6 and Example 4 explains how to find a basis for the column space of A . Here the answer is 1 0 1 0 1 0 1 1 0 - 1 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . In short, a 1 , a 2 , a 4 correspond to lead variables. a 3 , a 5 correspond to free variables. It is also clear that a 1 , a 2 , a 4 span and form a basis of the column space. If you don’t know why, read the above materials in detail. 2. Give an example of a (non-trivial) linear transformation from C [0 , 1] to R . Explain. Some examples are given in section 4.1, Problem 10 and parts of Problem 11. 3. True or False? Explain. Let L be a linear transformation from
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Unformatted text preview: V to W , and let T be a subspace of W . Then { v V | L ( v ) T } is a subspace of V . True. Verify directly from denitions of subspace and linear transformation. 4. Let E = 1 1 , 1 2 1 , -1 1 1 and F = 1 1 , 2 1 . Let L ( x ) be a linear transformation from R 3 to R 2 dened by L ( x ) = x 3 x 1 . Find the matrix representing L with respect to the ordered bases E and F . See supplement for Chap 4 for the situating when you are ONLY given the matrix representation of L in standard bases. Here L is explicitly given, so one just computes L 1 1 = 1 1 , then expand 1 1 in terms of the basis F . This gives the rst column of the matrix A = L E,F . One then proceed with L 1 2 1 and L -1 1 1 ....
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