# Confused about dimension of a range of a linear map

Suppose you have a linear map $$T:Mat_{2\times3}(R)\rightarrow Mat_{2\times2}(R)$$

Why is the range(T) 4 at most? I thought that this is a 2x2 matrix and that the maximum range is 2?

• What is the range of a linear map? The rank of a linear map is the dimension of its image. So, the rank of a linear map $T \colon V \to W$ is always less than or equal to $\dim(W)$. Feb 8 at 4:05
• The dimension of $\rm Mat_{2×2}(\Bbb R)$ is $4$. Feb 8 at 4:07
• There's another level of abstraction here. The vector spaces consist in matrices. As a matrix, your linear map would need to be $4×6.$ Feb 8 at 4:47

Why is the [dimension of] range(T) 4 at most? I thought that this is a 2x2 matrix and that the maximum range is 2?

$$\text{range}(T)$$ is not a $$2\times 2$$ matrix, but a SET of $$2\times 2$$ matrices. Precisely, it is a subspace of $$W=M_{2\times 2}(\mathbb R)$$. Its dimension has no connection with the maximum rank of the elements of $$W$$.

The dimension of the range of a linear application $$f: V\to W$$ is at most the minimum between the dimensions of $$V$$ and $$W$$. Since $$\dim(M_{n\times m}(\mathbb R))=nm$$, the dimension of $$T$$ is at most $$2\times 2=4$$.

If you choose a basis of $$V=M_{2\times 3}(\mathbb R)$$ and a basis $$M_{2\times 2}(\mathbb R)$$, then you can represent $$T$$ by a $$4\times 6$$ matrix (whose rank will be the dimension of $$\text{range}(T)$$).

• Oh, this clears up my confusion. Thank you so much. Feb 8 at 4:17

Range$$(T)$$ is a subspace of $$M_{2\times 2}(R)$$, which is a vector space of dimension $$4$$. Because the dimension of any subspace is less than or equal to its parent space, it follows that the largest possible dimension of Range$$(T)$$ is $$4$$.

You can verify that the dimension of $$M_{2\times 2}(R)$$ is $$4$$ by verifying that the list of matrices $$\begin{bmatrix}1 & 0 \\ 0& 0 \end{bmatrix}, \begin{bmatrix}0 & 1 \\ 0& 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \\ 1& 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0& 1 \end{bmatrix}$$ is linearly independent and spans $$M_{2\times 2}(R)$$.