If the linear transformation T : R2 → R3 is such that T(1, 0) = (2,3,1) and T(1,1) = (3, 0,2), thena)T(x, y) = (x +y, 2x + y, 3x - 3y)b)T(x, y) = (2x + y, 3x - 3y, x +y)c)T(x, y) = (2x - y, 3x + 3y, x - y)d)T(x, y) = (x - y, 2x - y , 3x + 3y)Correct answer is option 'B'. Can you explain this answer?

Question

If the linear transformation T : R2 &rarr; R3 is such that T(1, 0) = (2,3,1) and T(1,1) = (3, 0,2), thena)T(x, y) = (x +y, 2x + y, 3x - 3y)b)T(x, y) = (2x + y, 3x - 3y, x +y)c)T(x, y) = (2x - y, 3x + 3y, x - y)d)T(x, y) = (x - y, 2x - y , 3x + 3y)Correct answer is option 'B'. Can you explain this answer?

Answer

We are given that a linear transformation T : R³→ R³ defined by
T(1, 0) = (2, 3,1)
and T(l, 1) = (3, 0, 2)
We need to find the image of (x, y) under linear transformation T.
Let there exist scalars α and β such that
(x,y) = α(1,0) + β(1, 1)
or equivalently (x, y)= (α + β, β)
Comparing the components on both sides, we get
x = α + β and y = β
Solving for α and β, we get
α = x - y and β = y
Therefore,(x, y)= (x - y) (1, 0) + y(1, 1) Taking the image under linear transformation T. we get

implies T(x, y) = (2x + y, 3x - 3y, x+y).

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