Math325 8 qestions

Help me study for my Algebra class. I’m stuck and don’t understand.

1. Determine if the vectors   1 −2 3   ,   3 2 1  ,   5 6 −1   are linearly independent or linearly dependent. Please explain your answer. 2. Let A be the matrix   1 2 −3 1 0 2 −3 4 6  . For which vectors b does the equation Ax = b have a solution? 3. Does there exist a 3×3 matrix A that satisfies A2 = −I (where I denotes the identity matrix)? Hint: if A satisfies A2 = −I, what would det(A) be equal to? 4. If A is a 9 × 4 matrix, what is the smallest number of free variables the equation AT x = 0 can have? Please explain. 5. Find eigenvalues and eigenvectors of the matrix ? −14 12 −20 17? . 6. Give an example of a 2 × 2 matrix A and 2− dimensional vectors u and v such that u and v are orthogonal to each other, but the vectors Au and Av are not orthogonal to each other. 7. Apply Gram-Schmidt process to the vectors   1 −3 5   ,   2 2 1  . 8. Let A be an n × n matrix. Suppose that for some vector b the equation Ax = b has more than one solution. Explain why A is not an invertible matri

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