Math 21 1. Given A = 1 2 003 0 1 0, Long Quiz 2 - Page 2 of 6 2023/08/29 (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 2023/08/29 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. 2023/08/29 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. 2023/08/29 {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 2023/08/29 Is T is a linear transformation? If it is, prove it and find its standard matrix. If it is not, give a counterexample.

Fig: 1