Q1: What is not a property of a linear map $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$?

A1: $f( \mathbf{x} \cdot \mathbf{y} ) = f( \mathbf{x} ) \cdot f( \mathbf{y} ) $

A2: $f( \mathbf{x} + \mathbf{y} ) = f( \mathbf{x} ) + f( \mathbf{y} ) $

A3: $f( \lambda \mathbf{x}) = \lambda f( \mathbf{x} )$

A4: $f( \mathbf{x} + \lambda \mathbf{y} ) = f( \mathbf{x} ) + \lambda f( \mathbf{y} ) $

Q2: Let consider the vector space $X = \mathbb{R}$ and a map $f : X \rightarrow X$. Which of the statements is not correct?

A1: $f(x) = x^2$ is linear.

A2: $f(x) = x$ is linear.

A3: $f(x) = 2 x$ is linear.

A4: $f(x) = 3 x$ is linear.

A5: $f(x) = 0$ is linear.

Q3: Let consider the zero vector space $X = {0}$ and a map $f : X \rightarrow X$. Which of the statements is not correct?

A1: There is a map $f : X \rightarrow X$ that is not linear.

A2: $f(x) = x^2$ is linear.

A3: $f(x) = x$ is linear.

A4: $f(x) = 3 x$ is linear.

A5: $f(x) = 0$ is linear.