Unformatted text preview: T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS Linear Mathematics 2011 Quiz 1b  Solutions 0 2 0 1. Find the eigenvalues of the matrix 1 1 0. 0 1 1 a) 1, 1, 2 b) 1, 1, 2 c) 1, 1, 2 d) 1, 1, 2 Solution d) 2. The set S = t 1  t R is a subspace of R2 . 0 Which one of the following statements is not true? a) If b) If c) If d) If
x y x y x y x y S then S then S then S then x y 5x 5y x y + 0 1 S. S. S. 
x y 3 0 7 S. Solution a) 3. Suppose u and v are linearly independent vectors in R3 . Which one of the following is a basis for R3 ? a) {u, v, 2u  3v} b) {u, v, u v} c) {u, u + v, u  v} d) None of the above is a basis for R3 . Solution b) x 4. Let X = R2  x 0 . Which one of the following statements is true? y a) X is not closed under addition and is not closed under scalar multiplication. b) X is closed under addition, but not closed under scalar multiplication. c) X is closed under scalar multiplication, but not closed under addition. d) X is closed under addition and scalar multiplication. Solution b) Math 2061: Quiz 1b  Solutions S.B. 7/4/2011 Linear Mathematics
1 2 0 0 3 0 1 0 6 0 2 0 1 2 0 0 Quiz 1b  Solutions Page 2 5. Let X = a) b) c) d) 1 2 3 4 , , , . What is the dimension of Span (X)? Solution b) 6. Which one of the following is not a subspace of R3 ? s t a)  s, t R 0 b) c) d)
s 0 0 1 0 0 0 0 0 sR Solution c) 7. Suppose A is a 3 4 matrix and that the dimension of the null space of A is 2. What is the dimension of the column space of A? a) 1 b) 2 c) 3 d) 4 Solution b) 1 0 , 2 , 0 , . Which one of the following statements is 8. Consider the set U = 0 3 0 0 correct? a) U spans R3 but is not linearly independent. b) U is a basis of R3 . c) U is linearly independent but does not span R3 . d) U does not span R3 and U is not linearly independent. 1 0 0 1 Solution a) Math 2061: Quiz 1b  Solutions Page 2 Linear Mathematics Quiz 1b  Solutions Page 3 9. Suppose u, v, w, x are nonzero vectors in a vector space V , and that x is in Span {u, v, w} but x is not in Span {u, v}. Which one of the following statements is not correct? a) x must be a scalar multiple of w. b) Span {u, v, x} =Span {u, v, w}. c) w is not in Span {u, v}. d) {u, w} is a linearly independent set. Solution a) 10. Suppose A is a 4 6 matrix. Which one of the following is correct? a) Null(A) R4 and Col(A) R4 . b) Null(A) R6 and Col(A) R4 . c) Null(A) R4 and Col(A) R6 . d) Null(A) R6 and Col(A) R6 . Solution b) Questions 11 and 12 refer to the matrix A, which has reduced row echelon form J. 1 0 3 2 4 1 0 3 2 4 3 11 , A= 2 1 7 J = 0 1 1 1 3 . 1 3 0 5 5 0 0 0 0 0
1 3 0 2 3 5 2 1 0 11. Find a basis for Col(A). 1 0 3 2 , 1 , 7 , a) b) c) d)
1 0 0 1 0 0 1 2 1 , ,
4 3 0 0 1 1 , , , 0 1 0 0 1 0 0 1 3 , 3 1 0 , Solution d) 12. Find a basis for Null (A). a) b) c) d)
3 1 0 , 0 0 3 1 1 0 0 3 7 , 0 3 1 , 0 2 1 , 0 0 0 2 1 0 1 0 2 3 , 5 2 1 , 0 4 3 0 0 0 , ,
0 1 1 4 3 0 4 3 0 0 1 Solution b) Math 2061: Quiz 1b  Solutions Page 3 Linear Mathematics Quiz 1b  Solutions Page 4 13. The eigenvalues of the matrix M = 1 8 are 1 and 5. 1 3 Find a matrix which diagonalises M. a) 4 2 1 1 1 1 4 2 4 2 1 1 1 1 4 2 b) c) d) Solution c) 14. Let A be an n n diagonalisable matrix. Which one of the following is true? a) The columns of A are linearly independent. b) There exists a basis for Rn consisting of eigenvectors of A. c) All the entries in A that are not on the main diagonal are zero. d) A has no repeated eigenvalues. Solution b) 15. Define subsets S1 , S2 , S3 of the function space F as follows: S1 = {f F  f (3) = 0} S2 = {f F  f (x) = 3 for all x R} S3 = {f F  f (3) = f (3)} Which one of the following statements is true? a) b) c) d) S1 is a subspace of F, but S2 and S3 are not subspaces of F. S1 and S2 are subspaces of F, but S3 is not a subspace of F. S1 and S3 are subspaces of F, but S2 is not a subspace of F. None of these sets is a subspace of F. Solution c) Math 2061: Quiz 1b  Solutions Page 4 ...
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