Expert Answer to "Let T be a linear transformation from P2..."

glamrockqueen7

Answered question

2021-01-10

Let T be a linear transformation from

P_{2} into P_{2}

such that

T (1) = x, T (x) = 1 + x and T (x^{2}) = 1 + x + x^{2} .

Find

T (2 - 6 x + x^{2})

lamanocornudaW

Skilled2021-01-11Added 85 answers

Approach:
Let V and W be vector spaces. The function,

T : V \to W

is a linear transformation of V into W when the two properties below are thue for all u and v in V and for any scalar c.

T (u + v) = T (u) + T (v)

T (c u) = c T (u)

Calculation:
The value to be find is

T (2 - 6 x + x^{2}) .

2 - 6 x + x^{2} = c_{1} (1) + c_{2} (x) + c_{3} x^{2}

Compare the coefficient of same power.

c_{1} = 2

c_{2} = - 6

c_{3} = 1

Consider a linear transformation T such that,

v = c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n}

Then,

T (v) = T (c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n})

= c_{1} T (v_{1}) + c_{2} T (v_{2}) + \dots + c_{n} T (v_{n})

So,

T (2 - 6 x + x^{2}) = 2 T (1) - 6 T (x) + T (x^{2}) \dots (1)

Subtitute x for

T (1), (1 + x) f or T (x) and (1 + x + x^{2}) f or T (x^{2})

in equation (1).

T (2 - 6 x + x^{2} = 2 (x) - 6 (x + 1) + (1 + x + x^{2})

2 x - 6 x - 6 + 1 + x + x^{2}

= - 5 - 3 x + x^{2}

Therefore, the value of

T (2 - 6 x + x^{2}) i s - 5 - 3 x + x^{2} .

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