Find the characteristic polynomial of A

Reference no: EM132728594

Linear Algebra

1. (a) Complete the definition: Vectors (v1) ?, . . . , (vk) ? are linearly independent if...

(b) Suppose A is an m × n matrix whose columns are linearly independent.

What is the nullity of A?

2. Let A denote the matrix

A= [((-√3)/2&-1/2@1/2&-√3/2)]
Let T : R2 → R2 be the linear transformation given by T ( x) = A x. (a) (5 points) Describe T geometrically.

(b) Find the characteristic polynomial of A, and use it to find all eigen- values of A or to show that none exist. Explain why your answer makes sense geometrically.

3. (a) Compute the inverse of the matrix [(-4&0&5@-3&3&5@-1&2&2)]

(b) Find all solutions to the system of linear equations

-4x + 5z = -2
-3x - 3y + 5z = 3
-x + 2y + 2z = -1

4. Using Gaussian elimination, find all solutions to the following system of linear equations:
2x2 + 3x3 + 4x4 = 1
x1 - 3x2 + 4x3 + 5x4 = 2
-3x1 + 10x2 - 6x3 - 7x4 = -4

5. Let A denote the matrix
[(1&0&-2@0&5&0@-2&0&4)]
(a) (4 points) Find the eigenvalues of A.

(b) Find an orthonormal basis of R3 consisting of eigenvectors for A.

Reference no: EM132728594

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