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

A1: $f(\lambda x) = \lambda + f(x)$

A2: $f(x+y) = f(x) + f(y)$

A3: $f^{-1}$ exists.

A4: $f^{-1}$ is linear.

A5: $f^{-1}$ is bijective.

Q2: If there is a bijective linear map $f: U \rightarrow V$ between two subspaces $U, V \subseteq \mathbb{R}^n$, then $\mathrm{dim}(U) = \mathrm{dim}(V)$. Is this correct?

A1: Yes!

A2: No, there are counterexamples.

A3: One needs more information.

Q3: Let $U = { \mathbf{x} \in \mathbb{R}^2 \mid \langle \mathbf{x} , \binom{2}{1} \rangle = 0}$. Now, take another subspace $V \subseteq U$ with $\mathrm{dim}(V) = 1$. Which claim is correct?

A1: $V = U$.

A2: $\mathrm{dim}(U) = 2$.

A3: $\mathrm{dim}(U) = 3$.

A4: $V$ is not a subspace.

A5: $V = { 0 }$.