To find: The linear transformation (T_2T_1)(v) for an arbitrary vector v in V. The vectors {v_1,v_2} is vasis for the vector space V. Given: The linea

Question

To find:
The linear transformation

(T_{2} T_{1}) (v)

for an arbitrary vector v in V.
The vectors

{v_{1}, v_{2}}

is vasis for the vector space V.
Given:
The linear transformation with satisfying equations

T_{1} (v_{1}) = 3 v_{1} + v_{2},

T_{1} (v_{1}) = - 3 v_{1} + v_{2}, T_{2} (v_{1}) = - 5 v_{2},

and

T_{2} (v_{2}) = - v_{1} + 6 v_{2}

are given as

T_{1} : V \to V

and

T_{2} : V \to V .

Answer 1

Calculation:
To solve for

T_{2} (T_{1} (v))

find

T_{1} (v)

use properties of linear transformation.

T_{1} (v) = T_{1} (a v_{1} + b v_{2})

T_{1} (v) = a T_{1} (v_{1}) + b T_{1} (v_{2})

Substitute

3 v_{1} + v_{2}

for

T_{1} (v_{1})

and 0 for

T_{1} (v_{2})

in the above expression.

T_{1} (v) = a (3 v_{1} + v_{2}) + b (0)

= a (3 v_{1} + v_{2})

Substitute

a (3 v_{1} + v_{2})

for

T_{1} (v)

to find

T_{2} (T_{1} (v))

and it is written as,

T_{2} (T_{1} (v)) = T_{2} (a (3 v_{1} + v_{2}))

T_{2} (T_{1} (v)) = 3 a T_{2} (v_{1}) + a T_{2} (v_{2})

Substitute

- 5 v_{2}

for

T_{2} (v_{1})

and

- v_{1} + 6 v_{2}

for

T_{2} (v_{2})

in the above expression.

T_{2} (T_{1} (v)) = 3 a (- 5 v_{2}) + a (- v_{1} + 6 v_{2})

- 15 a v_{2} - a v_{1} + 6 a v_{2}

= - a v_{1} - 9 a v_{2}

Therefore, the linear transformation

(T_{2} T_{1}) (v)

is

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To find: The linear transformation (T_2T_1)(v) for an arbitrary vector v in V. The vectors {v_1,v_2} is vasis for the vector space V. Given: The linea

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