Q1: Let $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be injective. Which is statement is always correct?

A1: If $(v_1,v_2, \ldots, v_k)$ is linearly independent in $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is linearly independent in $W$.

A2: $f$ is also surjective.

A3: $f$ is also invertible.

A4: If $\mathcal{F} = (v_1,v_2, \ldots, v_k)$ is a family with $\mathrm{Span}(\mathcal{F}) = V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ also spans $W$.

Q2: Let $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be surjective. Which is statement is always correct?

A1: If $(v_1,v_2, \ldots, v_k)$ is linearly independent in $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is linearly independent in $W$.

A2: $f$ is also injective.

A3: $f$ is also invertible.

A4: If $\mathcal{F} = (v_1,v_2, \ldots, v_k)$ is a family with $\mathrm{Span}(\mathcal{F}) = V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ also spans $W$.

Q3: Let $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be bijective. Which is statement is not correct in general?

A1: If $(v_1,v_2, \ldots, v_k)$ is a basis of $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is a basis of $W$.

A2: $f^{-1}$ is linear.

A3: $f$ is also invertible.

A4: $f(0) = 0$

A5: $f^{-1} \circ f$ is not a linear map.

A6: $f$ is an isomorphism.

Q4: Let $f: V \rightarrow W$ be an isomorphism between two $\mathbb{F}$-vector spaces $V$ and $W$. What is always correct?

A1: $\dim(V) = \dim(W)$

A2: $\dim(V) < \dim(W)$

A3: $\dim(V) > \dim(W)$