Exercise · ZW-LINEAR-TRANSFORMATION-009 · ★★★ Advanced
Let \(P\) be the space of polynomials. Which of these functions from \(P\) to \(P\) or to \(\mathbb{R}\) is linear?
Choose one option.
Show answer
- A. \(T(p)=p'\), the derivative
Explanation
The derivative satisfies \((p+q)'=p'+q'\) and \((cp)'=cp'\). The others fail: \(p(0)+1\) sends \(0\) to \(1\), \((p+q)^2\ne p^2+q^2\), and the absolute value is not additive (\(|1+(-1)|\ne|1|+|-1|\)).
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-002 Which of these functions from ℝ^2 to ℝ^2 is linear?
- 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-008 ZW-LINEAR-TRANSFORMATION-010 →