Math 21

1. Given A

=

1 2

003

0 1 0,

Long Quiz 2 - Page 2 of 6

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(a) (5 points) Use Gaussian elimination / Gauss-Jordan row reduction to find the in-

verse A-¹. Make sure to clearly indicate each elementary row operation that you

use in the process.

(b) (4 points) Write A as a product of elementary matrices. Math 21

Long Quiz 2 - Page 3 of 6

20 0

15 0-3

()

895 7

430 6

(a) (5 points) Compute the determinant of B.

2. Given B

(b) (2 points) Using your answer above, justify whether or not the sequence of vectors

(0000))

7

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is a basis for R4. Math 21

a

{ ( 1 ) : a

(a) (3 points) Prove that W is a subspace of M2×2(R).

3. Suppose W

=

Long Quiz 2 - Page 4 of 6

R}.

: a, b, c ER

(b) (2 points) Consider the basis B

=

coordinate vector of w

=

(69). ( )· (69))

(²3) with respect to B.

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for W. Find the Math 21

4. (5 points) Consider the subspace U of R₁[X] given by

4

Long Quiz 2 - Page 5 of 6

U

=

3c=0}

Find a basis for U and determine dim(U). Make sure to prove that your proposed basis

is indeed a basis for U.

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{a+bX+cX² + dX³ + eXª : a +2c = 0 and b Math 21

Long Quiz 2 - Page 6 of 6

5. (4 points) Let T : R³ → R² be the transformation defined by

3x 2y 4z

7 (4) = (30 - 22 +4.)

Ty

5x – 9z

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Is T is a linear transformation? If it is, prove it and find its standard matrix. If it is

not, give a counterexample.