Unformatted text preview: FIRST EXAM-B Instructions: Begin each of the eight problems on a new page in your answer book.
Show your work. 1. [18 Pts.] For each matrix below, determine whether its columns form a linearly independent set. Give reasons for your
answers. (Make as few calculations as possible.) 2 —l 3 —5 0 2 —l 0 4
a. 8 —4 b. —6 2 4 c. —6 3 0 —7
—4 l 9 —7 —4 4 l 0 —l 2. 18 For each matrix in problem (1), determine if the columns of the matrix span R3. Give reasons for your answers.
(Make as few calculations as possible.) 3. 10 Suppose A, B, C, and X are all invertible n X n matrices, and A(X + B) 2 CA. Solve this equation for X. Show
your work. 4. 10 Let T: R2 —> R2 be the linear transformation that rotates points through an angle of 37r / 2 radians about the origin.
Find the standard matrix of T. 5. 12 Suppose the columns of a 5 X 5 matrix A are linearly independent. What can you say about equations of the form
Ax = b? (Say as much as you can about Ax = b.) Justify your answer. 6. 12 Suppose A is a 6 X 6 matrix such that the equation Ax = b has no solution for some b in R6. What can you say
about the transformation x »—> Ax? (Say as much as you can about x »—> Ax.) Justify your answer. 7. 10 Let C be an m X 71 matrix and D be an n X p matrix. Give the deﬁnition of CD. Then, suppose that the ﬁrst column
of D is the sum of columns 2 and 3 of D. What can you say about the columns of the product CD. Say as much as you
can. Justify your answer. 8.  Let A be an m X 71 matrix with columns a1, . .. ,an (with m and 71 any integers greater than 1). Explain why the
columns of A are linearly independent if and only if the equations Ax = 0 has only the trivial solution. Your explanation
should show that you know the deﬁnition of Ax and the deﬁnition of linear independence. (If you have trouble with this
problem, begin by writing what it means for the vectors a1, . . . ,an to be linearly independent.) ...
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