math 21 1 given a 1 2 003 0 1 0 long quiz 2 page 2 of 6 2023 08 29 a 5

Question

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.