Let T: P2 → P3 be the transformation that maps a polynomial p(t) into the polynomial (t-2)p(t). a. Find the image of p(t) = 2-t + t^2
b. Show that T is a linear transformation.
c. Find the matrix for T relative to the bases (1, t, t^2) and (1,t, t^2,t^3).

(a) Applying T to p(t) = 2 - t + t ² gives

T ( p(t) ) = (t - 2) (2 - t + t ²) = -4 + 4t - 3t ² + t ³

(b) T is a linear transformation if for any p(t) and q(t) in P₂ and complex scalars a and b, the image of any linear combination of p and q is equal to the linear combination of the images of p and q. In other words,

T ( a p(t) + b q(t) ) = a T ( p(t) ) + b T ( q(t) )

Let

p(t) = α₀ + α₁ t + α₂ t ²

q(t) = β₀ + β₁ t + β₂ t ²

Compute the images of p and q :

T ( p(t) ) = (t - 2) (α₀ + α₁ t + α₂ t ²)

… = -2α₀ + (α₀ - 2α₁) t + (α₁ - 2α₂) t ² + α₂ t ³

Similarly,

T ( q(t) ) = -2β₀ + (β₀ - 2β₁) t + (β₁ - 2β₂) t ² + β₂ t ³

Then

a T ( p(t) ) + b T ( q(t) ) = a (-2α₀ + (α₀ - 2α₁) t + (α₁ - 2α₂) t ² + α₂ t ³) + b (-2β₀ + (β₀ - 2β₁) t + (β₁ - 2β₂) t ² + β₂ t ³)

… = c₀ + c₁ t + c₂ t ² + c₃ t ³

where

c₀ = -2 (a α₀ + b β₀)

c₁ = a (α₀ - 2α₁) + b (β₀ - 2β₁)

c₂ = a (α₁ - 2α₂) + b (β₁ - 2β₂)

c₃ = a α₂ + b β₂

Computing the image of a p(t) + b q(t) would give the same result; just multiply it by t - 2 and expand. This establishes that T is indeed linear.

(c) Find the image of each vector in the basis for P₂ :

T (1) = (t - 2) × 1 = t - 2

T (t ) = (t - 2) t = t ² - 2t

T (t ²) = (t - 2) t ² = t ³ - 2t ²

Then

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕

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