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

  • University of Toronto
  • MAT 223
  • Test Prep
  • jxscarlettgao
  • 16
  • 100% (1) 1 out of 1 people found this document helpful

This preview shows page 6 - 16 out of 16 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Introduction to the Theory of Computation
The document you are viewing contains questions related to this textbook.
Chapter 2 / Exercise 2.53
Introduction to the Theory of Computation
Sipser
Expert Verified
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
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Introduction to the Theory of Computation
The document you are viewing contains questions related to this textbook.
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
THIS PAGE LEFT INTENTIONALLY BLANK. If any work on this page is to be graded, indicate this CLEARLY. 15
THIS PAGE LEFT INTENTIONALLY BLANK. If any work on this page is to be graded, indicate this CLEARLY. 16

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture