MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 4
NAME: Section:(Circle one) A(1 : 10) B(2 : 40)
Show ALL your work CAREFULLY.
Let T:R3→R2be a linear transformation defined by
T(x1, x2, x3) = (2x1−3x2+x3,4x3−x1).
(a) Find the standard matrix Aof Tso that T(~x) = A~x.
The matrix Ahas columns T(~e1), T (~e2), T (~e3). Since T(~e1) = T(1,0,0) = (2,−1),T(~e2) =
T(0,1,0) = (−3,0),T(~e3) = T(0,0,1) = (1,4), it follows that the matrix Ais given by
A=2−3 1
−104.
(b) Find all vectors ~x such that T(~x) = ~
0.
Since T(~x) = A~x, the vectors ~x such that T(~x) = ~
0are precisely the solutions to the
homogeneous equation A~x =~
0. Now,
A=2−3 1
−104∼1−3 5
−104∼1−3 5
0−3 9∼1 0 −4
0−3 9 ∼1 0 −4
0 1 −3.
The solutions to A~x =~
0are
~x =
x1
x2
x3
=
4x3
3x3
x3
=x3
4
3
1
where x3is the parameter.
(c) Is Tone to one? Explain.
No. From (b), the homogeneous equation A~x =~
0has non-trivial solutions so Tcannot
be one-to-one.
Date: February 1, 2013.
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