1. Determine if the vectors1−23 ,321,56−1 are linearly independent orlinearly dependent. Please explain your answer.2. Let A be the matrix1 2 −31 0 2−3 4 6. For which vectors b does the equationAx = b have a solution?3. Does there exist a 3×3 matrix A that satisfies A2 = −I (where I denotesthe 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 theequation 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 andv such that u and v are orthogonal to each other, but the vectors Au andAv are not orthogonal to each other.7. Apply Gram-Schmidt process to the vectors1−35 ,221.8. Let A be an n × n matrix. Suppose that for some vector b the equationAx = b has more than one solution. Explain why A is not an invertiblematri 100

1. Determine if the vectors

6 are linearly independent or

linearly dependent. Please explain your answer.

1 2 -3

2. Let A be the matrix 1 0 2 . For which vectors b does the equation

-3 4 6

Ac = b have a solution?

3. Does there exist a 3 x 3 matrix A that satisfies A2 =-1 (where I denotes

the identity matrix)?

Hint: if A satisfies A² =-1, what would det(A) be equal to?

4. If A is a 9 x 4 matrix, what is the smallest number of free variables the

equation A?r= 0can have? Please explain.

-14 12]

5. Find eigenvalues and eigenvectors of the matrix

-20 17

6. Give an example of a 2 x 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

00

8. Let A be an nxn matrix. Suppose that for some vector b the equation

Ac =

b has more than one solution. Explain why A is not an invertible

matrix.