Exercise · ZW-LINEAR-TRANSFORMATION-006 · ★★☆ Intermediate
A linear transformation satisfies \(T(1,1)=(3,1)\) and \(T(1,-1)=(1,5)\). What is \(T(1,0)\)?
Choose one option.
Show answer
- A. (2, 3)
Explanation
Since \((1,0)=\tfrac12(1,1)+\tfrac12(1,-1)\), \(T(1,0)=\tfrac12\big[(3,1)+(1,5)\big]=(2,3)\).
Review the concept: Linear transformation
A linear transformation \(T\) is a function between vector spaces that preserves addition and scalar multiplication.
\[T(u+v)=T(u)+T(v),\quad T(cu)=cT(u)\]
More practice problems: linear transformation
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- ZW-LINEAR-TRANSFORMATION-005 A linear transformation T satisfies T(1,0)=(2,1) and T(0,1)=(-1,3). Compute T(3,2).
- ZW-LINEAR-TRANSFORMATION-007 A linear transformation satisfies T(u)=(1,2) and T(v)=(3,-1). Compute T(2u-v).
- ZW-LINEAR-TRANSFORMATION-008 Which of these geometric operations on the plane is NOT a linear transformation?
- ZW-LINEAR-TRANSFORMATION-009 Let P be the space of polynomials. Which of these functions from P to P or to ℝ is linear?
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