Linear Algebra

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Moderate

Affine is not Linear

LINALG-XISN0B

Let $A$ be an $n\times n$ real matrix and $\vec{x}\in\mathbb{R}^n$ a fixed vector and define:

$$T:\mathbb{R}^n\rightarrow\mathbb{R}^n$$

...by:

$$T(\vec{v})=A\vec{v}+\vec{x}$$

...for each:

$$\vec{v}\in\mathbb{R}^n$$

Then, $T$ is a linear transformation:

A

For all $\vec{x}\in\mathbb{R}^n$

B

For all $\vec{x}\in\mathbb{R}^n$ provided that $A$ is invertible

C

Only when $\vec{x}=\vec{0}$

D

Only for $\vec{x}\in Col(A)$