Image of the Zero matrix

Question

I'm learning some introductory linear algebra and am confused about the zero matrix's image. Is that just the point $\langle 0,0,0 \rangle$ / zero vector?

Answer 1

Let $A$ be a real $m\times n$ matrix. Then $A$ defines a linear mapping $\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$, where the domain is $\mathbb{R}^{n}$ and the codomain is $\mathbb{R}^{m}$.

The image of $A$ is the set of all vectors in $\mathbb{R}^{m}$ that we get when we input any vector in $\mathbb{R}^{n}$ into the mapping, i.e.

$$\mathrm{im} (A)=\left\{A\mathbf{x}\ \vert\ \mathbf{x}\in \mathbb{R}^{n}\right\}.$$

In the case of the $m\times n$ zero matrix $\mathbf{0}$, we have $$\mathrm{im} (\mathbf{0})=\left\{\mathbf{0}\mathbf{x}\ \vert\ \mathbf{x}\in \mathbb{R}^{n}\right\}.$$

Since $\mathbf{0}\mathbf{x}=\mathbf{0}\in \mathbb{R}^{m}$ for any $\mathbf{x}\in \mathbb{R}^{n}$, then the image of the zero map is indeed the trivial subspace $\left\{\mathbf{0}\right\}.$

Answer 2

Let's calculate!

$$ \left[\matrix{0}\right] \left[\matrix{x_1}\right] = \left[ \matrix{0 \cdot x_1} \right] = \left[ \matrix{0} \right]$$

$$ \left[\matrix{0 & 0 \\ 0&0}\right] \left[\matrix{x_1\\x_2}\right] = \left[ \matrix{0 \cdot x_1 + 0 \cdot x_2 \\ 0 \cdot x_1 + 0 \cdot x_2} \right] = \left[ \matrix{0\\0} \right] $$

$$ \left[\matrix{0 & 0 & 0 \\ 0&0&0}\right] \left[\matrix{x_1\\x_2\\x_3}\right] = \left[ \matrix{0 \cdot x_1 + 0 \cdot x_2 + 0\cdot x_3 \\ 0 \cdot x_1 + 0 \cdot x_2 + 0\cdot x_3} \right] = \left[ \matrix{0\\0\\0} \right] $$

$$ \left[\matrix{0 & 0 \\ 0&0\\0&0}\right] \left[\matrix{x_1\\x_2}\right] = \left[ \matrix{0 \cdot x_1 + 0 \cdot x_2 \\ 0 \cdot x_1 + 0 \cdot x_2 \\0 \cdot x_1 + 0 \cdot x_2} \right] = \left[ \matrix{0\\0\\0} \right] $$

and so on and so forth.