Transformations and their dimensions

Question

If $X$ is a linear transformation from $Y$ to $\mathbb{R}^{2×2}$ and $\mathrm{ker}(X) = \{0\}$, then which of the following statements about $\mathrm{dim}(Y)$ is necessarily true?

• $\mathrm{dim}(Y) \leq 4$
• $\mathrm{dim}(Y) \geq 4$
• $\mathrm{dim}(Y) = 4$

If the solution is necessarily true, the solutions would be $A$,$B$,$C$ given that $\mathrm{dim}(Y)$ will be $4$ necessarily?

Answer 1

Hint: The is a famous theorem connecting the dimensions of kernel, image and vector space.

Answer 2

Hint:

Suppose $X: Y \to \mathbb{R}^{2 \times 2}$ is a linear transformation with $\text{ker}(X) = \{\mathbf{0}\}$.

Then since $\text{ker}(X) = \{\mathbf{0}\}$, it follows that $X$ is [...]?

The dimension of $\mathbb{R}^{2 \times 2}$ is $4$. Therefore, since $X$ is [...], it follows that $\dim(Y) \quad ?\quad 4$.