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Kuwait University, Department of Mathematics Math 111, Linear Algebra 11 January, 2012, 2-4pm. Final Exam. Answer all questions. Calculators and mobile phones are NOT allowed. ( 1. (3+2 pts) Let A =

3 −2

6 2

)

( and B =

3 −3 1 k

) .

(a) Express the columns of AB as linear combinations of the columns of A. (b) Find all values of k for which AB = BA. 2. (3+2 pts) (a) Find parametric equations for the line of intersection of the two planes x + y + z = 3 and x − 2y + z = 0. (b) Let U = (a, b, c) and V = (x, y, z) be two vectors in ℜ3 . Show that (ax + by + cz)2 ≤ (a2 + b2 + c2 )(x2 + y 2 + z 2 ).

3. (3+2 pts) (a) Show that the vector U = (1, 2, 1, 1) does not belong to span{(1, 2, 3, 0), (2, 1, 6, 0)}. (b) Let X = (10, 0, c), Y = (1, 0, 5) and Z = (0, −1, 0)} be three vectors in ℜ3 . Determine all values of c for which the vector X is orthogonal to the plane containing the vectors Y and Z. 4. (3+2 pts) (a) Does there exist a ∈ ℜ such that S = {(1, 1, 1), (2, 2, 2), (3, a, 4)} is a basis for ℜ3 . (b) Let A be a 3 × 3 matrix such that the system AX = 0 has the following two vectors X1 = (1, 0, 1)T and X2 = (0, 1, 1)T as solutions. Find one more nontrivial solution to the system AX = 0.    5. (3+1+2+1 pts) Let A =  

1 0 0 0

0 3 0 0

4 9 0 0

1 3 2 3

   . 

(a) Find a basis for the null space of A. (b) Find a basis for the column space of A containing only columns of A. (c) Determine whether the vector X = (0, 3, 2, 3)T belongs to the column space of A. (d) Determine the rank and nullity of A.    6. (7 pts) Let A =  

2 0 0 1

0 2 0 0

0 0 1 0

0 0 0 1

   . 

If possible, find a non-singular matrix P and a diagonal matrix D such that P −1 AP = D. 7. (6 pts) Answer each of the following as True or False (justify your answer). (a) Let A be an n × n matrix. If 0 is an eigenvalue of A then rank(A) = n. (b) If A and B are two n × n matrices with |A| = 1 and |B| = −1, then the columns of AB span ℜn . (c) If U and V are two orthogonal vectors in ℜn , then ||U + V ||2 − ||U − V ||2 = 0. (d) W = {(x, y, z)|2x + 3y + z = 0, 3x + 2y − z = 0 and x − y + 7z = 0} is a subspace of ℜ3 .

µ 1. (3+2 pts) Let A =

3 −2

6 2

¶

µ and B =

3 1

−3 k

¶ .

(a) Express the columns of AB as linear combinations of the columns of A. µ ¶ µ ¶ µ ¶ 3 6 15 col1 (AB) = A col1 (B) = 3 +1 = . −2 2 −4 µ ¶ µ ¶ µ ¶ 3 6 −9 + 6k col2 (AB) = A col2 (B) = −3 +k = . −2 2 6 + 2k (b) Find all values of k for which AB = BA. µ ¶ µ 15 −9 + 6k 15 AB = and BA = −4 6 + 2k 3 − 2k i.e., when k = 27 .

12 6 + 2k

¶ . AB = BA when 3 − 2k = −4 and −9 + 6k = 12

2. (3+2 pts) (a) Find parametric equations for the line of intersection of the two planes x + y + z = 3 and x − 2y + z = 0. µ ¶ µ ¶ µ ¶ µ ¶ 1 1 1 3 1 1 1 3 1 1 1 3 1 0 1 2 → → → : Parametric equations for 1 −2 1 0   0 −3 0 −3 0 1 0 1 0 1 0 1  2−t x 1  , where t ∈