Linear Algebra

Math 2318 Final Exam

Submit your work via the drop box for the Final Exam in Angel. Each problem is worth 7 points.

Given the following system of equations

3x1  5x2 − 4x3  b1 − 3x1 − 2x2  4x3  b2

6x1  x2 − 8x3  b3

1.Write this as matrix equation and identify A, x and b

Ax  b

2. Corresponding to matrix A there is a linear transformation T, what is this transformation and this transformation goes from what space to what space.

3. What is the determinant for matrix A and what does this tell you about matrix A being invertible or not?

4. What is a basis for the Null Space of A, what is the rank of the Null Space and what does this tell you about the linear transformation being one-to-one?

5. What is the dimension of the Column Space of A and what does this tell you about the linear transformation being onto or not?

6. Given the following transformation which is called the Lorentz Transformation

L2  5/4 3/4

3/4 3/4

If you pick the corner vertices of a square centered at the origin

1

1 ,

1

−1 ,

−1

1 ,

−1

−1

and L2 maps this square from 2 into 2 then what is the shape of the square under the transformation? Where did each of these vertices get mapped to?

7. In the previous problem what is the area of the original square and what is the area of the new object under the transformation?

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8. What is the determinant of matrix L2 and is matrix L2 invertible. If it is invertible then find the inverse?

9. If L2 is invertible what does this tell you about the dimension of the Null Space of matrix and is the corresponding linear transformation one-to-one or not?

10. Given the following matrix find the eigenvalues and the eigenvectors

A  3 5

−2 −4

11. Given the following matrix find the eigenvalues and the eigenvectors

A  3 4

−5 −5

12. Find a basis for all of the vectors in 2 that lie on the line y  4x

13. What is the "standard basis" for P2 and how would you write the following vector from P2 in terms of that basis

pt  3 − 4t  5t2

14. Is the following collection of vectors also a basis for P2 and why? If it is a basis what is the change of coordinate matrix from this basis to the standard basis.

B  1 − 3t2,2  t − 5t2,1  2t

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