Published on 31 Jan 2019

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MT Q S

İ S. Y

M ,

Surname, Name:

Q . §. M O

Let T:R2ÏR3be the linear transformation deﬁned by

(x1, x2)7Ï (x1+x2, x2, x1−x2)

and S:R3ÏR3be the linear transformation deﬁned by

(x1, x2, x3)7Ï (x1+x2+x3, x1−x2−x3,−x1−x2−x3).

Find the standard matrix of S◦T.

A

Note that the domain of Tis R2and the codomain is R3. We need to consider

the 2×2identity matrix. We need to ﬁnd T(e1), T(e2):

T(e1) = (1,0,1), T(e2) = (1,1,−1) or

T(e1) =

1

0

1

, T(e2) =

1

1

−1

.

Therefore, the standard matrix Aof Tis given as

T(x) = A·xwhere A =

1 1

0 1

1−1

.

Note that the domain of Sis R3and the codomain is R3. We need to consider

the 3×3identity matrix. We need to ﬁnd S(e1), S(e2), S(e3):

S(e1) = (1,1,−1), S(e2) = (1,−1,−1), S(e3) = (1,−1,−1), or

S(e1) =

1

1

1

, S(e2) =

1

−1

−1

, S(e3) =

−1

−1

−1

.

Therefore, the standard matrix Bof Sis given as

S(x) = B·xwhere B =

1 1 −1

1−1−1

1−1−1

.

We get the standard matrix of S◦Tas

BA =

1 1 −1

1−1−1

1−1−1

1 1

0 1

1−1

=

0 2

0 2

0 2

.

1