PracticeExamFinal

# PracticeExamFinal - ~v = " 1 2 # is an eigenvector for...

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MATH 214 Practice Final Exam December 9, 2007 1. Find all values of the parameters a and b for which the system of linear equations ± ± ± ± ± x + 2 y = a 4 x + by = 5 ± ± ± ± ± (a) has no solutions; (b) has a unique solution; (c) has inﬁnitely many solutions.

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2. Find a basis for the kernel and the image of the matrix A = 1 0 1 0 2 1 3 - 1 0 1 1 - 1 . Describe the image and the kernel geometrically (line, plane, etc., inside. ..).
3. Let ~v 1 = " 2 3 # and ~v 2 = " 1 2 # be the basis of R 2 . Find the coordinate vector of ~x = " - 4 11 # .

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4. Let T : U 2 × 2 -→ U 2 × 2 be the linear transformation of upper triangular 2 × 2 matrices: T : M 7-→ M " 1 2 0 1 # - " 1 2 0 1 # M. (a) Find the matrix of T under the standard basis B = ±" 1 0 0 0 # , " 0 1 0 0 # , " 0 0 0 1 #! . (b) Find the determinant of T .
5. Compute the orthogonal projection of the vector ~x = 5 5 5 onto the plane spanned by the vectors ~v 1 = 3 0 4 and ~v 2 = 3 2 4 .

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6. Compute the determinant A = 1 - 1 0 1 2 1 5 - 1 - 1 2 - 2 0 3 - 1 - 2 3 .
7. Compute the eigenvalues and ﬁnd an eigenbasis of the matrix A = " 6 - 3 3 - 4 # .

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8. Suppose that the vector

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Unformatted text preview: ~v = " 1 2 # is an eigenvector for a matrix A and that the corresponding eigenvalue is 2. Let B = A 4-3 A . Compute B~v . 9. Let A be the following matrix. Determine whether it is diagonalizable or not. If it is, ﬁnd an invertible S and a diagonal D such that D = S-1 AS ; if it is not, explain why. (a) A = 1 1 1 1 1 1 1 1 1 (b) A = 1 1 1 0 1 0 0 1 0 10. Let A be a 2 × 2 matrix. Assume that " 1 1 # is an eigenvector of A with eigenvalue λ 1 =-2 and that " 1 2 # is an eigenvector of A with eigenvalue λ 2 = 3. (a) Compute A ; (b) Find a closed formula for A n where n is a positive integer. 11. Let L : R 2 × 2-→ R 2 × 2 be the linear transformation L ( A ) = A + A T , for any A ∈ R 2 × 2 . Find the eigenvalues and eigenmatrices of L . Is L diagonalizable?...
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## This note was uploaded on 04/03/2008 for the course MATH 214 taught by Professor Conger during the Fall '08 term at University of Michigan.

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PracticeExamFinal - ~v = " 1 2 # is an eigenvector for...

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