Exercise · ZW-LINEAR-TRANSFORMATION-002 · ★☆☆ Basic
Which of these functions from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) is linear?
Choose one option.
Show answer
- D. \(T(x,y)=(x+y,\,x-y)\)
Explanation
Only \((x+y,x-y)\) is a sum of multiples of \(x\) and \(y\), with no constants or powers. The others fail: \(x+1\) moves the origin, and \(x^2\) and \(xy\) are not additive.
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
- ZW-LINEAR-TRANSFORMATION-001 Let T(x,y)=(2x,3y). What is T(1,1)?
- ZW-LINEAR-TRANSFORMATION-003 If T is a linear transformation, what is T(0)?
- ZW-LINEAR-TRANSFORMATION-004 Let T(x,y)=(x+y, x-y). Compute T(2,3).
- 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-006 A linear transformation satisfies T(1,1)=(3,1) and T(1,-1)=(1,5). What is T(1,0)?
- 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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