Let be the linear transformation satisfying Tv1 4 7 Tv2 1

Let

be the linear transformation satisfying

T(v₁) = (4, 7),    T(v₂) = (1, 1),

where

v₁ = (1, 1) and v₂ = (1, 1).

Find

T(x₁, x₂)

for an arbitrary vector

(x₁, x₂)

T(x₁, x₂) =

What is

T(6, 2)?

T(6, 2) =

Solution

Let A =

1

1

1

0

1

-1

0

1

We will reduce A to its RREF as under:

Add -1 times the 1st row to the 2nd row

Multiply the 2nd row by -1/2

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1

0

½

½

0

1

½

-1/2

Thus, we have (1,0)^T = ½(1,1)^T +1/2(1,-1)^T and (0,1)^T = ½(1,1)^T -1/2(1,-1)^T

Then (x₁, x₂) = x₁(1,0)+x₂(0,1) = x₁/2(1,1)^T +x₁/2(1,-1)^T +x₂/2(1,1)^T - x₂/2 (1,-1)^T. Therefore, T((x₁,x₂)^T) = T[x₁/2(1,1)^T +x₁/2(1,-1)^T +x₂/2(1,1)^T - x₂/2 (1,-1)^T] = (x₁/2)T( (1,1)^T)+(x₁/2) T((1,-1)^T) +(x₂/2) T((1,1)^T ) - (x₂/2) T((1,-1)^T) = (x₁/2)(4,7)^T +(x₁/2)(-1,1)^T + (x₂/2)(4,7)^T - (x₂/2)(-1,1)^T = (x₁/2)(3,8)^T +(x₂/2)(5,6)^T =   ½( 3x₁+5x₂ ,8x₁ +6x₂)^T. On substituting x₁ = 6 and x₂ = -2, we get T((6,-2)^T ) = ½( 18-10, 48-12)^T = ½(8,36)^T = (4,18)^T