# W x 1 x 2 x 3 x 4 r 4 x 1 x 3 x 4 of r 4 4 marks 6 2

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Chapter 2 / Exercise 2.53
Introduction to the Theory of Computation
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W = 4 ? [4 marks] x 1 x 2 R 4 | x 1 + x 3 = x 4 R 4 | x 1 + x 3 = x 4 6
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Chapter 2 / Exercise 2.53
Introduction to the Theory of Computation
Sipser
Expert Verified
2. (c) Is the set S = 1 1 0 1 , 0 1 1 1 , - 1 1 2 1 a basis for the subspace W in part (b)? [4 marks] 7
3 (a). State the Rank-Nullity Theorem. [2 marks] 3. (b) Let A = (i) What are rank( A ) and nullity( A )?. [2 marks] (ii) Find a basis for the row space of A . [2 marks] 2 2 0 0 1 0 1 - 2 2 1 1 - 2 . 8
(iii) Find a basis for the column space of A . [2 marks] (iv) Find a basis for the nullspace of A . [2 marks] 9
4. (a) Let A = 0 c - c - 1 2 1 c - c c . Use determinants to find all values of c such that A is invertible. [5 marks] 10
4. (b) Let A = a b c d e f g h i and suppose det( A ) = 3. Find det(2 B - 1 ) where B = 2 d 2 e 2 f - a + g - b + h - c + i 3 g 3 h 3 i . [5 marks] 11
5. Let P be the parallelogram with vertices (2 , 3) , ( - 1 , 4) , (5 , 7), and (2 , 8). (a) Sketch P and find its area using determinants. [4 marks] 12
(b) Let T : R 2 R 2 be the linear transformation defined by T ( x ) = A x , where A = 1 4 2 5 . Find the area of T ( P ). [4 marks] 13
6. Let A be an 3 × 3 matrix with columns a 1 , a 2 , a 3 . If rank( A ) = 3, prove that the set { A a 1 , A a 2 , A a 3 } is a basis for R 3 . [5 marks] 14