Expert answers to "Let L ::P2⇒P3 be a linear transformation for..."

komanijuuxw

Answered question

2022-04-29

Let L :

: P_{2} \Rightarrow P_{3}

be a linear transformation for which we
know that

L : (1) = 1, L (t) = t^{2}, L (t^{2}) = t^{3} = t .

(a) Find

L (2 t^{2} - 5 t = 3) .

(b) Find

L (a t^{2} - b t + c) .

tswe0uk

Beginner2022-04-30Added 19 answers

Step 1
Given,

L : P^{2} \Rightarrow P^{3}

be a linear transformation for which we
know that

L (1) = 1, L (t) = t^{2}, L (t^{2}) = t^{3} = t .

We use the properties of a linear transformation
L(X+Y)=L(x)+L(Y)
L(aX)=aL(X)
Part a)
Since L is a linear transformation, we write

L (2 t^{2} - 5 t + 3) = L (2 t^{2}) = L (- 5 t) = L (3)

= 2 L (t^{2}) - 5 L (t) + 3 L (1)

= 2 (t^{3} = t) - 5 t^{2} + 1

= 2 t^{3} - 5 t^{2} + 2 t + 1

Part b)
Since L is a linear transformation, we write

L (a t^{2} + b t + 3) = L (a b^{2}) + L (b t) + L (c)

= a L (t^{2}) = b L (t) = c L (1)

= a (t^{2} + t) + b t^{2} + c

= a t^{3} + b t^{2} + a t + c

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