Winter 2007 - Wickramasekera's Class - Practice Exam 2

# Winter 2007 - Wickramasekera's Class - Practice Exam 2 -...

This preview shows pages 1–3. Sign up to view the full content.

MATH 20F: LINEAR ALGEBRA, WINTER 2007 MIDTERM 2 PRACTICE PROBLEMS Note: This is not a practice exam; you do not have to be able to do these problems all in 50 minutes. (1) Find the inverse of the matrix A below or show that A is not invertible. A = 1 2 3 2 5 3 1 0 8 . (2) Determine if the set { x 3 - 1 , - x 2 - x - 2 , 2 x 3 + x 2 + 1 } of polynomials is linearly independent. (3) The matrices A = 1 - 3 - 1 3 1 - 2 6 1 - 5 - 6 3 - 9 - 2 8 7 1 - 3 0 2 5 and B = 1 - 3 0 2 5 0 0 1 - 1 4 0 0 0 0 0 0 0 0 0 0 are row equivalent. (a) What is the rank of A ? (b) Find bases for Col ( A ) and Nul ( A ). (4) The maximal number of linearly independent rows of a 5 × 6 matrix A is 3. What is the dimension of its null space? (5) Suppose A = " 5 4 2 3 # and B = " 7 3 2 1 # . Define maps L : R 2 R 2 and T : R 2 R 2 by L ( x ) = A x and T ( x ) = B x . Is there a linear map 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
S : R 2 R 2 such that S ( T ( x )) = L ( x ) for every x in R 2 ? If so, find an explicit formula for S ( x ) and if not, explain why not. (6) Repeat problem 5 with A = " 4 2 8 4 # and B = " 6 3 2 1 # . (7) Let M be the set whose only element is the sun. Define addition and scalar multiplication on M by sun + sun = sun and c (sun) = sun. Is M a vector space under these operations? (8) Show that the set of vectors of the form ( a, b, c ) , where a, b, c are real numbers and b = a + c, is a subspace of R 3 . Find a basis for and di- mension of this subspace. (9) Consider the bases B = { u 1 , u 2 , u 3 } , and C = { v 1 , v 2 , v 3 } for R 3 where u 1 = (3 , 0 , 3), u 2 = (3 , - 2 , 1), u 3 = ( - 1 , - 6 , 1), v 1 = (6 , 6 , 0), v 2 = (2 , 6 , - 4) and v 3 = (2 , 3 , - 7) .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern