# Injective and Surjective Linear Maps Examples 2

Recall from the Injective and Surjective Linear Maps page that a linear map $T : V \to W$ is said to be **injective** if:

- $T(u) = T(v)$ implies that $u = v$.

- $\mathrm{null} (T) = \{ 0 \}$.

Furthermore, the linear map $T : V \to W$ is said to be surjective if:

- If for every $w \in W$ there exists a $v \in V$ such that $T(v) = w$.

- $\mathrm{range} (T) = W$.

We will now look at some more examples regarding injective/surjective linear maps.

## Example 1

**Recall that that if $V$ and $W$ are vector spaces then the set of all linear maps from $V$ to $W$, $\mathcal L(V, W)$ with addition defined by $(S + T)(v) = S(v) + T(v)$ and scalar multiplication defined by $(aT)(v) = aT(v)$ for all $S, T \in \mathcal L (V, W)$ and for all $a \in \mathbb{F}$, is a vector space. Suppose that $2 ≤ \mathrm{dim} (V) ≤ \mathrm{dim} (W)$. Prove that then the subset $\{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is NOT a subspace of $\mathcal L (V, W)$.**

To show that $U = \{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is not a subspace of $\mathcal L (V, W)$ – we must either show that $0 \not \in \mathcal L (V, W)$ or show that $\{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is not closed under addition or scalar multiplication.

Note that the zero linear map $0 : V \to W$ defined by $0(v) = 0$ which maps every vector $v \in V$ to $0 \in W$ is not injective because $0(v) = 0(w)$ does not imply that $v = w$. Therefore the zero map is not injective and:

(1)

So we have thus far failed to prove that $\{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is not a subspace of $\mathcal L(V, W)$.

We will now check to see if $\{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is closed under addition. Let $S, T \in \{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$.

Let $\{ v_1, v_2, …, v_n \}$ be a basis of $V$ and let $\{ w_1, w_2, …, w_m \}$ be a basis of $W$. We note that $2 ≤ n ≤ m$. Define the linear map $S$ by:

(2)

(3)

Note that $S$ is not injective. To show this, suppose that $u, v \in V$. Then $u = a_1v_1 + a_2v_2 + … + a_nv_n$ and $v = b_1v_2 + b_2v_2 + … + b_nv_n$ for some set of scalars $a_1, a_2, …, a_n, b_1, b_2, …, b_n \in \mathbb{F}$, and so:

(4)

Note that the equation above does not imply that $u = v$. For example, if $u = a_1v_1 + a_2v_2 + … + a_nv_n$ and $v = b_1v_1 + a_2v_2 + … + a_nv_n$ where $a_1 \neq b_1$ then $S(u) = S(v)$ but clearly $u \neq v$.

We also note that $T$ is not injective. To show this, suppose that $u, v \in V$. Then once again, $u = a_1v_1 + a_2v_2 + … + a_nv_n$ and $v = b_1v_1 + b_2v_2 + … + b_nv_n$ for some set of scalars $a_1, a_2, …, a_n, b_1, b_2, …, b_n \in \mathbb{F}$, and so:

(5)

Note that the equation above does not imply that $u = v$. For example, if $u = a_1v_1 + a_2v_2 + … + a_nv_n$ and $v = a_1v_1 + b_2v_2 + a_3v_3 + … + a_nv_n$ where $a_2 \neq b_2$ then $S(u) = S(v)$ but clearly $u \neq v$.

Now consider the linear map $(S + T)(v) = S(v) + T(v)$ for all $v \in V$. For $u$ and $v$ defined as the linear combinations of the basis vectors previously, we have that:

(6)

if we subtract both sides of the equations above, we have that:

(7)

Since $\{ w_1, w_2, …, w_m \}$ is a basis – it is also a linearly independent set. Note that $n ≤ m$ (from earlier), and so the set of vectors $\{ w_1, w_2, …, w_n \}$ as a subset of the prescribed basis of $W$ is also linearly independent. This implies that:

(8)

Therefore $u = v$, so $S + T$ is injective. Therefore, $(S + T) \not \in \{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ so $\{ T \in \mathcal L (V, W) : T \: \mathrm{is \: not \: injective} \}$ is not closed under addition and hence is not a subspace of $\mathcal L (V, W)$.

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