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Let $V$ be a vector space, $v, u ∈ V$, and let $T_1: V \to V$ and $T_2: V \to V$ be linear transformations such that $T_1(v) = 6 v + 5 u$, $T_1(u) = -4 v - 3 u$, $T_2(v) = 2 v - 6 u$, and $T_2(u) = -5 v + 2 u$. Find the images of $v$ and $u$ under the composite of $T_1$ and $T_2$.

$(T_2 \circ T_1)(v) =$ ?

and

$(T_2 \circ T_1)(u) =$ ?

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closed as off-topic by Jonas Meyer, Grigory M, Fundamental, Ivo Terek, Sujaan Kunalan Jan 12 at 2:30

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1 Answer 1

up vote 2 down vote accepted

It's just a question of using the definition of composition. For instance:

$$(T_2\circ T_1)(v)=T_2(T_1(v))$$

But you know $T_1(v) = 6v+5u$ and hence you've got

$$(T_2\circ T_1)(v)=T_2(6v+5u)$$

However you also know that $T_2$ is linear, so that you can write

$$(T_2\circ T_1)(v)=6T_2(v)+5T_2(u)$$

And again you know both $T_2(v)$ and $T_2(u)$ so you've got

$$(T_2\circ T_1)(v)=6(2v-6u) +5(-5v+2u)$$

Which is simply

$$(T_2\circ T_1)(v)= -13v-2u$$

Well, too much detailed, but I think it will guide you well on this kind of exercise. Try the other one yourself, it's the way to learn math: looking some examples and doing by yourself.

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