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Material Type: Exam; Professor: Li; Class: LINEAR ALGEBRA; Subject: Mathematics ; University: Oklahoma State University - Stillwater; Term: Fall 2008;
Typology: Exams
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INSTRUCTOR: WEIPING LI
Print Name and Student #
(1) (10pts) Find a basis for the orthogonal complement W ⊥^ in R^4 of the subspace W = sp{(1, 2 , 2 , 1), (3, 4 , 2 , 3)}.
(2) (10pts) Find the projection of (1, 2 , 1) on the subspace W = sp{(3, 1 , 2), (1, 0 , 1)} in R^3.
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(3) (15pts) Find an orthogonal basis from the basis {(1, 0 , 1), (0, 1 , 2), (2, 1 , 0)} for R^3 by using the Gram-Schmidt process.
(4) (10pts) Verify the matrix A = (^17)
(^) is orthogonal, and find its inverse.
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(7) (15pts) Find (i) the rank of A and the nullity of A, (ii) a basis for the row space, (iii) a basis
for the column space of A =
(8) (10pts) Enlarge the given independent set {(1, 1 , −1), (1, 2 , −2)} in R^3 to a basis for the entire space R^3
LINEAR ALGEBRA 5
(9) (15pts) Let T be a linear transformation such that T (− 1 , 2) = (1, − 1 , 0) and T (3, −5) = (0, 1 , −1). (i) Find T (x, y).
(ii) Using (i), compute T (− 1 , 1).
(10) (10pts) Find the coordinate vector of the polynomial 3x^3 − x^2 + 3x + 20 relative to the
ordered basis B 1 = {(x − 1)^3 , (x − 1)^2 , (x − 1), 1 }.
LINEAR ALGEBRA 7
(13) (10pts) Let A be a 5 × 5 matrix with det A = 4. Find the following.
(i) det(AT^ + AT^ )
(ii) det(AT^ · A)
(iii) det(A−^3 ).
(14) (10pts) Find the x 2 ONLY by using Cramer’s rule for
5 x 1 − 2 x 2 + x 3 = 1 3 x 1 + 2x 2 = 3 x 1 + x 2 − x 3 = 0.
(15) (10pts) Find the characteristic polynomial of
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(16) (10pts) Let A =
. The characteristic polynomial of A is given by
(λ − 1)(λ + 3)^2. (i) Find the algebraic multiplicities of λ 1 = 1 and λ 2 = −3.
(ii) Find the geometric multiplicities of λ 1 = 1 and λ 2 = −3.
(i) Is the matrix A diagonalizable?
(17) (15pts) A matrix A =
(^) has eigenvalues 2, 1 , −1. Compute Ak.
