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I S. Y

M ,

Q . THE MATRIX OF A LINEAR TRANSFORMATION

Deﬁne the linear transformation T:R3ÏR3so that

x1

x2

x3

7Ï

x1+x2+x3

x1−x2+x3

x1−x2−x3

.

a.) Find the standard matrix of T.

b.) Is Tone-to-one? Explain.

c.) Is Tonto? Explain.

d.) If there is any, ﬁnd a vector ⃗v such that T(⃗v) = ⃗

bwhere ⃗

b=

2

2

1

.

1

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Q . MATRIX OPERATIONS

Deﬁne the linear transformations T:R3ÏR3and S:R3ÏR3so that

T

x1

x2

x3

=

2x1−2x2+x3

3x2−x3

x3

and S

x1

x2

x3

=

−2x1−2x2+x3

x2−3x3

x3

.

a.) Find the standard matrix of S◦T.

b.) Find the standard matrix of T◦S.

c.) Find, if there is any, a vector ⃗v such that (S◦T)(⃗v) = ⃗

bwhere ⃗

b=

1

1

0

.

2

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