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176 Chapter Six: Linear TYansformationns
We shall encounter other common properties of similar matri-
ces in Chapter Eight.
Exercises 6.2
1. Which of the following functions are linear transforma-
tions?
(a) Ti : R 3 —> R where Ti([2:1X2X3]) = \Jx\ + x\ + x§;
T
(b) T 2 : M m, n{F) - M n , m (F) where T 2(A) = A ;
(c) T 3 : M n(F) ->• F where T 3 (^) = det(4).
2. If T is a linear transformation, prove that T(—v) = —T(y)
for all vectors v.
3. Let I be a fixed line in the xy-plane passing through the
origin O. If P is any point in the plane, denote by P' the
mirror image of P in the line /. Prove that the assignment
2
OP —> OP' determines a linear operator on R . (This is
called reflection in the line I).
4. A linear transformation T : R —> R is defined by
Xl
( \ Xi — X2 — %3 ~ £4
T{ 2xi + x 2 - X3
X3
%2 - %3 + %4
\X4 /
Find the matrix that represents T with respect to the standard
3
bases of R 4 and R .
5. A function T : P^(R) —>• P^(R) is defined by the rule
/
T(f) = xf" — 2xf + . Show that T is a linear operator and
find the matrix that represents T with respect to the standard
basis of P4.(R).
6. Find the matrix which represents the reflection in Exercise
2
3 with respect to the standard ordered basis of R , given that
the angle between the positive x-direction and the line I is 4>.