Math 221: Matrix Algebra Midterm 1 - May 23, 2012
Instructions: There are ﬁve questions and 100 points on this exam. You will need only a pen or pencil and eraser; nothing else is permitted. Unless otherwise indicated, write your ﬁnal answers clearly in complete sentences; failure to do so will cost points. Point values indicated for each question are estimates and subject to change. (1) (12 points) Suppose that T : Rn → Rn which is not one-to-one, and let A denote the standard matrix of T . Indicate whether the following are true or false by writing the complete word True or False (you will lose points for simply writing T or F). (a) A is a square matrix. (b) The columns of A are linearly dependent. (c) A has a pivot in every column. (d) T is not onto.
(2) (16 points) For each of the following mappings, write linear if the mapping is a linear transformation, and otherwise write not linear. You do not need to justify your answers. (a) T : R2 → R3 deﬁned by T x y x+y = x2 0
(b) T : R2 → R3 deﬁned by T x y x+y = x+1 0
(c) T : R → R3 deﬁned by
x T (x) = −2x 0
(d) T : Rn → Rm deﬁned by T (x) = 0.
(3) (24 points) Suppose that a linear system has a coeﬃcient matrix A whose reduced echelon form REF (A) is 1 −1 0 −1 0 0 0 1 −1 0 REF (A) = 0 0 0 0 1 0 0 0 0 0
(a) Express the solution set for the homogeneous linear system Ax = 0 in parametric vector form.
(b) Suppose that a1 , a2 , a3 , a4 , and a5 are the columns of A, so that A= a1 a2 a3 a4 a5 Express the zero vector 0 ∈ R4 as a linear combination of the columns of A in which not all the coeﬃcients are zero.
(c) Do the columns of A span R4 ? Justify your answer.
(4) (24 points) Suppose that T : R2 → R3 is a linear transformation such that T (e1 − 2e2 ) = 3e1 − e3 Let A be the standard matrix of T . (a) How many rows and columns does A have? T (−e1 + e2 ) = −2e2
(b) Find the matrix A.
(c) Is T one-to-one? Is T...