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Material Type: Exam; Class: Introductory Linear Algebra; Subject: Mathematics; University: Washington State University; Term: Fall 2007;
Typology: Exams
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(a) Is p =
(^) in Col A? Why or why not?
(b) Is q =
(^) in Nul A? Why or why not?
(a) Use determinants to decide if A is invertible or not.
(b) Are the columns of A linearly independent? Justify.
(c) Is the transformation x → Ax one-to-one?
(d) Does the transformation x → Ax map R^4 onto R^4?
(e) What is the dimension of the null space of A?
(a) Find the image of x =
(^) under this transformation.
(b) Based on the result obtained above (and without doing any further calculations), find an eigenvalue and a corresponding eigenvector of A.
(a) Is λ = −2 an eigenvalue of A? Why or why not? If yes, find an associated eigenvector.
(b) Is x =
an eigenvector of A? Why or why not? If yes, find the corresponding eigenvalue.
(a) Do the columns of A span R^3? Explain.
(b) Do the columns of A form a basis for R^3? Explain.
T (e 1 ) =
h
(^) , and T (e 2 ) =
k 0
. Determine all the values of the parameters h and k for which T is one-to-one.
(^) has eigenvalues 2, 2 , and −1.
(a) Give the characteristic polynomial of A. You need not simplify your answer.
(b) Determine a basis for the eigenspace corresponding to the eigenvalue λ = 2.
(^) , u 2 =
(^) , u 3 =
(^) , and x =
(a) Determine if the set { u 1 , u 2 , u 3 } is orthogonal.
(b) Is { u 1 , u 2 , u 3 } a basis for R^3? If yes, express x as a linear combination of u 1 , u 2 , and u 3.
(a) If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
(b) Every linearly independent set in Rn^ is an orthogonal set.
(c) For square matrices A and B, det(A + B) = det A + det B.
(d) A number c is an eigenvalue of the matrix A if and only if the equation (A − cI)x = 0 has a solution.
(e) If the equation Ax = b is inconsistent for some b in Rn, then Ax = 0 has only the trivial solution.
