Which of the following functions T: ℝ² → ℝ² is a linear transformation?
AT(x, y) = (x + 1, y)
BT(x, y) = (2x, 3y)
CT(x, y) = (x², y)
DT(x, y) = (x + y², x - y)
T(x, y) = (2x, 3y) is linear: scaling and adding inputs produces scaled and added outputs. Option A fails because T(0, 0) = (1, 0) ≠ (0, 0) — a linear transformation must send the zero vector to zero. Option C fails because x² is not a linear operation. Option D fails because y² is not linear.
Question 2 True / False
If T is a linear transformation, then T must send the zero vector to the zero vector.
TTrue
FFalse
Answer: True
This follows directly from linearity. Setting c = 0 in T(cv) = cT(v) gives T(0) = T(0·v) = 0·T(v) = 0. This is a useful quick test: if a function doesn't send the zero vector to zero, it cannot be linear — which is exactly why T(x) = x + 1 fails despite looking almost linear.
Question 3 Short Answer
What does it mean to say a linear transformation 'preserves structure', and which two operations does it preserve?
Think about your answer, then reveal below.
Model answer: A linear transformation preserves vector addition and scalar multiplication. 'Preserving structure' means you get the same result whether you add or scale vectors first and then transform, or transform first and then add or scale.
Formally: T(u + v) = T(u) + T(v) and T(cv) = cT(v). This makes linear transformations algebraic homomorphisms — they respect the vector space structure. A non-linear function like T(x) = x² scrambles this structure: T(u + v) ≠ T(u) + T(v) in general.