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Winter 2007 - Wickramasekera's Class - Practice Final Exam

# Winter 2007 - Wickramasekera's Class - Practice Final Exam...

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Unformatted text preview: MATH 20F: LINEAR ALGEBRA, WINTER 2007 PRACTICE PROBLEMS FOR THE FINAL EXAM Note: This is not a practice exam; you don’t have to be able to do these problems all in 3 hrs. Some of these problems are longer than any of the problems on the actual exam. (1) The maximal number of linearly independent rows of a 5 × 6 matrix A is 3. What is the dimension of its null space? (2) Suppose A is a 5 × 9 matrix and b a vector in R 5 . If the rank of A is 2 and the rank of the augmented matrix [ A b ] is 3, can the system A x = b be consistent? (3) Suppose A is a 5 × 4 matrix of rank 1, and that the three vectors u = [3- 2 1- 1] T , v = [2 2 1 2] T and w = [- 1 4 2 3] T satisfy the equation A x = . (a) Show that { u , v , w } is a basis for the null space of A . (b) Is the vector [1 1 1 1] T in the null space of A ? (4) (a) Let P 2 be the vector space of polynomials in x of degree at most 2. Show that the polynomials 1, x- 1 and ( x- 1) 2 form a basis for P 2 . (b) Use (a) to show that if p ( x ) is a polynomial of degree ≤ 2, then there exist numbers a , b and c such that the following partial fractions expression holds for x 6 = 1: p ( x ) ( x- 1) 3 = a ( x- 1) 3 + b ( x- 1) 2 + c ( x- 1) . (Note: You are not asked to compute a , b and c .) (5) For the matrix A below, answer the following questions without doing any row reduction. (a) What is the rank of A ? (b) Find a basis for the column space of A . (c) Show that the two vectors u = [- π 0 0 0 0 1] T and v = [- 2 1 0 0 0 0] T belong to the null space of A . Do they span the null space? (d) Find a vector p such that A p = [1 1 2 3] T . (e) Give an infinite family of solutions to the system A x = [1 1 2 3] T . A = 1 2 e ln4 7 π 1 2 π √ 2 1 π 2 4 e + π √ 2 + ln4 8 2 π 3 6 2 e + π 2ln4 + √ 2 15 3 π . 1 (6) Let T : R 2 → R 3 be a linear transformation. Suppose T ([2- 1] T ) = [1 0 1] T and T ([2 3] T ) = [- 1 2 0] T . (a) Find the standard matrix A of T. (b) Find T ([3- 5] T ) . (c) Is T one-to-one? Is it onto? (7) (a) Show that the transformation T : R 3 → R 2 given by T ([ x y z ] T ) = [2 x + 3 y- 7 z 0] T is linear, and find a matrix A such that T ( x ) = A x for all x in R 3 . (b) What is the image of the plane 2 x + 3 y- 7 z = 0 under T ? (8) Give the matrix for the linear transformation which rotates vectors in R 2 counterclockwise by angle θ. (9) Let a be a real number and T : R 2 → R 2 be the transformation given by T ([ x y ] T ) = [ ax (5- a ) y ] T . Let D be the region in R 2 enclosed by the unit circle { ( x,y ) : x 2 + y 2 = 1 } . If the area of the image T ( D ) of D under T is 6 π , find all possible values of a . (10) Find all 2 × 2 matrices B such that AB = BA where A = 1 2 0 3 ....
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Winter 2007 - Wickramasekera's Class - Practice Final Exam...

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