Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.

Express $\mathbf{0}_n=\mathbf{0}_n + \mathbf{0}_n$ or $\mathbf{0}_n=0\cdot \mathbf{0}_n$.

Calculate $T(\mathbf{0}_n)$ using step 1 and the definition of linear transformation.

I will give three proofs.

Proof 1.

Since $\mathbf{0}_n=\mathbf{0}_n + \mathbf{0}_n$, we have
\[ T( \mathbf{0}_n)=T( \mathbf{0}_n + \mathbf{0}_n)=T( \mathbf{0}_n)+T( \mathbf{0}_n),\]
where the second equality follows since $T$ is a linear transformation.

Subtracting $T( \mathbf{0}_n)$ from both sides of the equality, we obtain $\mathbf{0}_m=T(\mathbf{0}_n)$.
Note that $\mathbf{0}_m=T( \mathbf{0}_n)-T( \mathbf{0}_n)$ since $T(\mathbf{0}_n)$ is a vector in $\mathbb{R}^m$.

Proof 2.

Observe that we have $0\cdot \mathbf{0}_n=\mathbf{0}_n$. (This is a scalar multiplication of the scalar $0$ and the vector $\mathbf{0}_n$.

Here we used the one of the properties of the linear transformation $T$ in the second equality.

Proof 3.

Note that $\mathbf{0}_n=\mathbf{0}_n – \mathbf{0}_n$.
Thus we have
\[ T( \mathbf{0}_n)=T( \mathbf{0}_n – \mathbf{0}_n)=T( \mathbf{0}_n)-T( \mathbf{0}_n)=\mathbf{0}_m,\]
where we used the linearity of $T$ in the second equality.
In the last equality, note that the vector $T(\mathbf{0}_n)$ is $m$-dimensional vector.

Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation?
Determine whether the function $T:\R^2 \to \R^3$ defined by
\[T\left(\, \begin{bmatrix}
x \\
y
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_+y \\
x+1 \\
3y
\end{bmatrix}\]
is a linear transformation.
Solution.
The […]

Linear Transformation and a Basis of the Vector Space $\R^3$
Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.
Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$.
Show […]

Determine a Value of Linear Transformation From $\R^3$ to $\R^2$
Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
\[ T\left(\, \begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}\,\right) =\begin{bmatrix}
1 \\
2
\end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}
0 \\
1 \\
1
[…]

Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
0
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\]
where $\mathbf{e}_1, […]

Find an Orthonormal Basis of the Range of a Linear Transformation
Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\]
Find an orthonormal basis of the range of $T$.
(The Ohio […]

A Linear Transformation from Vector Space over Rational Numbers to itself
Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.
(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the […]

Null Space, Nullity, Range, Rank of a Projection Linear Transformation
Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) […]

[…] that any linear transformation maps the zero vector to the zero vector. (See A linear transformation maps the zero vector to the zero vector for a proof of this […]

[…] To show that $mathbf{x}, Tmathbf{x}, T^2mathbf{x}$ form a basis, it suffices to prove that these vectors are linearly independent since the dimension of the vector space $R^3$ is three and any three linearly independent vectors form a basis in such a vector space. (For a different proof, see the post A linear transformation maps the zero vector to the zero vector.) […]

## 3 Responses

[…] that any linear transformation maps the zero vector to the zero vector. (See A linear transformation maps the zero vector to the zero vector for a proof of this […]

[…] To show that $mathbf{x}, Tmathbf{x}, T^2mathbf{x}$ form a basis, it suffices to prove that these vectors are linearly independent since the dimension of the vector space $R^3$ is three and any three linearly independent vectors form a basis in such a vector space. (For a different proof, see the post A linear transformation maps the zero vector to the zero vector.) […]

[…] Recall that every linear transformation must map the zero vector to the zero vector. […]