Linear Algebra - Linear Transformations

Question

Consider the basis S = {v1, v2, v3} for R3, where v1 = (1, 2, 1); v2 = (2, 9, 0), and v3 = (3, 3, 4) and let T : R3-->R2 be the linear transformation for which

T(v1) = (1, 0), T(v2) = (1, 1); T(v3) = (0, 1)

Find a formula for T(x1, x2, x3), and use that formula to finnd T(7, 13, 7)

Answer 1

Let e₁ = (1, 0, 0), e₂ = (0, 1,0) and e₃ = (0,0,1) basis in R₃. Let express this bases as linear combination of'

v₁ = (1, 2, 1); v₂ = (2, 9, 0) and v₃ = (3, 3, 4);

e₁ = av₁ + bv₂ + cv₃ | e₂ = dv₁ + ev₂ + fe_{3 |} e₃ = kv₁ + lv₂ + mv₃

a + 2b +3c = 1 a = - 36 | d + 2e + 3f = 0 d = 8 | k + 2l + 3m = 0 k = 21

2a + 9b + 3c = 0 b = 5 | 2d + 9e + 3f = 1 e = - 1 | 2k + 9l + 3m = 0 l = - 3

a + 4c = 0 c = 9 | d + 4f = 0 f = - 2 | k + 4m = 1 m = - 5

e₁ = -36v₁ + 5v₂ + 9v_3; e₂ = 8v₁ - v₂ - 2v₃

T(e₁) = - 36(1, 0) + 5(1,1) + 9(0,1) = (- 31, 14)

T(e₂) = 8(1,0) - (1,1) - 2(0,1) = (7,- 3)

T(e₃) = 21(1,0) - 3(1,1) - 5(0,1) = (18, - 8)

T(x₁, x₂, x₃) = [(- 31 14) (7 - 3) (18 - 8)][x₁ x₂ x₃] = [(- 31x₁ + 7x₂ + 18x₃) (14x₁ - 3x₂ - 8x₃)]

T is 2X3 matrix.

At last:

T(7, 13, 7) = [(- 31·7 + 7·13 + 18·7) (14·7 - 3·13 - 8·7)] = [0 3]

Answer 2

To find how a linear transformation acts on an arbitrary vector, given how the transformation acts on a basis, we have to construct to standard basis for the space. That is, we have to find a how to express the vectors (1,0,0), (0,1,0) and (0,0,1) as a linear combination of the vectors (1,2,1), (2,9,0) and (3,3,4). Then, since we know that T is a linear transformation, we can use this linear combination to express T(x₁, x₂, x₃) for any vector (x₁,x₂,x₃).

For example, if I wanted to know what the value of T is for the input vector (7,14,7) I can take the fact that T(v₁)=T(1,2,1)=(1,0) and multiply by 7 to get T(7,14,7)=T(7v₁)=7T(v₁)=7(1,0)=(7,0). Similarly, I could find the value of T for the vector (7,16,6) = 2v₁+v₂+v₃ be taking the linear combination 2(1,0)+(1,1)+(0,1)=(3,2).

If we can find a linear combination for the ordinary basis (1,0,0), (0,1,0) and (0,0,1) then we will have a simple formula for T(x₁,x₂,x₃) namely we will know that T(x₁,x₂,x₃)=x₁T(1,0,0)+x₂T(0,1,0)+x₃T(0,0,1).