MT210 QUIZ 5
İLKER S. YÜCE
MARCH 15, 2011
Surname, Name:
QUESTION 1. §1.9 THE MATRIX OF A LINEAR TRANSFORMATION
Let T:R3ÏR3be the linear transformation defined by
T(x1, x2, x3) = (x1, x1+x2, x1+x2+x3).
Find the standard matrix of T. Is Tone to one? Is Tonto?
ANSWER
Note that the domain is R3and the codomain is R3. We need to consider the
3×3identity matrix. We need to find T(e1), T(e2), T (e3):
T(e1) = T((1,0,0)) = (1,1,1), T(e2) = T(0,1,0) = (0,1,1), T (e3) = (0,0,1), or
T(e1) =
1
1
1
, T(e2) =
0
1
1
, T(e3) =
0
0
1
.
Therefore, the standard matrix A of T is given as
T(x) = A·xwhe re A =
1 0 0
1 1 0
1 1 1
.
Note that we have
A=
1 0 0
1 1 0
1 1 1
∼
100
010
001
.
Therefore, the homogenous system Ax=0has only the trivial solution which
implies that (YES) T IS ONE-TO-ONE. Since the reduced echelon form of A
has three pivot positions, (YES) T IS ONTO.
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