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Sample Exam 2

# Sample Exam 2 - Math 322 Sample Exam 2 Partial credit is...

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Math 322, Sample Exam # 2 Partial credit is possible, but you must show all work. Name: I hereby testify that this is individual work. Signed: 1. (a) Find the eigenvalues and eigenvectors of the matrix A = 1 - 2 - 3 2 (b) By starting with the definition A -→ x = λ -→ x , explain why λ n is an eigenvalue for A n with the same eigenvector. (c) Write A in the form P D P - 1 , where D is diagonal. Do the same for A - 3 . What do you observe?

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2 2. Find the characteristic equation and eigenvalues of the matrix A = 1 1 0 - 1 0 1 - 2 1 0
3 3. (a) Find the column subspace corresponding to the transformation T x y z = x - z x + y + z (b) Find the null subspace corresponding to the transformation T of part (a).

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4 4. (a) Find the kernel corresponding to the transformation T x y z = x - y + z x + 2 y + z (b) Solve the equation T -→ x = -→ b using the decomposition -→ x = -→ x h + -→ x p where -→ x h is an element of the kernel of T . Here -→ b = 1 - 1
5 5. (a) Find a basis for the subspace spanned by the vectors -→ u 1 = 1 - 2 3 , -→ u 2 = 1 - 1 0 , -→ u 3 = 2 0 - 1 , -→ u 4 = 0 - 2 - 1 . (b) Express the vector

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Sample Exam 2 - Math 322 Sample Exam 2 Partial credit is...

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