Null Spaces and Ranges

Null spaces and ranges of linear transformation

The most important concepts we encountered for matrices were secretly built from linear transformations.

For example, the null space and range of a matrix. We have the following definitions.

Definition. Let $T: V \to W$ be a linear transformation. The null space of $T$ is the set $\mathcal Z(T) := \lbrace v \in V \mid T(v) = 0 \rbrace$ and the range of $T$ is the set $\mathcal R(T) := \lbrace w \in W \mid w = T(v) \rbrace$

The nullity of $T$ is $\dim \mathcal Z(T)$ and the rank of $T$ is $\dim \mathcal R(T)$ .

Lemma. The null space $\mathcal Z(T)$ is a subspace of $V$ and the range $\mathcal R(T)$ is a subspace of $W$ .

Proof. (Expand to view)

We check that each are closed under linear combinations of two vectors.

Let $v_1,v_2 \in \mathcal Z (T)$ . Then, $T(c_1v_1 + c_2v_2) = c_1T(v_1) + c_2T(v_2) = 0$ so $c_1v_1 + c_2v_2 \in \mathcal Z(T)$ .

Let $w_1,w_2 \in \mathcal R(T)$ . We can therefore write $w_i = T(v_i)$ . Now $c_1w_1 + c_2w_2 = c_1T(v_1) + c_2T(v_2) = T(c_1v_1+c_2v_2)$ so $c_1w_1+c_2w_2 \in \mathcal R(T)$ . ■

Proposition. Assume that $\dim V, \dim W < \infty$ and that $A$ is a matrix representation for $T$ in some bases for $V$ and $W$ . Then, $\mathcal Z(T) \cong \mathcal Z(A)$ and $\mathcal R(T) \cong \mathcal R(A)$

Proof. (Expand to view)

Let’s choose bases $v_1,\ldots,v_n$ for $V$ and $w_1,\ldots,w_m$ for $W$ and have $A$ be the matrix representation of $T$ in these bases.

Take $v \in \mathcal Z(T)$ and write it in the basis $v = \sum_{i=1}^n c_i v_i$ Denote $\mathbf{c} := \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix}$ Let’s write out $A \mathbf{c}$ $(A \mathbf{c})_j = \sum_{j=1}^m A_{ji} c_i$ Applying $T$ directly to $v$ gives $T(v) = \sum_{i=1}^n c_i T(v_i) = \sum_{i=1}^n c_i \left(\sum_{j=1}^m A_{ji} w_j \right) = \sum_{j=1}^m \left(\sum_{i=1}^n A_{ji} c_i \right) w_j$ If $T(v) = 0$ , then, as $w_1,\ldots,w_j$ is a basis, we see that $\sum_{i=1}^n A_{ji} c_i = 0$ for all $1 \leq j \leq m$ . In other words, $\mathbf{c} \in \mathcal Z(A)$ .

In the other direction, if $\mathbf{c} \in \mathcal Z(A)$ , then $T(v) = 0$ . Thus, $\begin{aligned} \mathcal Z(T) & \to \mathcal Z(A) \\ v & \mapsto \mathbf{c} \end{aligned}$ is an isomorphism.

Next we turn to the ranges. Given $w \in W$ , we write it in the vector representation for the basis $w_1,\ldots,w_m$ $w = \sum_{j=1}^m d_j w_j$ We will check that $\begin{aligned} \mathcal R(T) & \to \mathcal R(A) \\ w & \mapsto \mathbf{d} \end{aligned}$ is an isomorphism. From the computation above, $T(v) = \sum_{i=1}^n c_i T(v_i) = \sum_{i=1}^n c_i \left(\sum_{j=1}^m A_{ji} w_j \right) = \sum_{j=1}^m \left(\sum_{i=1}^n A_{ji} c_i \right) w_j$ we see that $T(v)$ in vector representation has $A\mathbf{c}$ as coefficients. Thus, $w = T(v)$ if and only if $\mathbf{d} = A \mathbf{c}$ so the above map is an isomorphism. ■

The previous statement tells us that we can use our familiar tools for matrices to understand null spaces and ranges for linear transformations once we choose a basis.

As an application of this principle, we can quickly deduce the Rank-Nullity Theorem for linear transformations.

Theorem. (Rank-Nullity) Let $T: V \to W$ be a linear transformation with $\dim V, \dim W < \infty$ . Then, $\operatorname{rank} T + \operatorname{nullity} T = \dim V.$

Proof. (Expand to view)

Since $\mathcal Z(T) \cong \mathcal Z(A)$ and $\mathcal R(T) \cong \mathcal R(A)$ , we have $\operatorname{nullity} T = \dim \mathcal Z(T) = \dim \mathcal Z(A) = \operatorname{nullity} A$ and $\operatorname{rank} T = \dim \mathcal R(T) = \dim \mathcal R(A) = \operatorname{rank} A$ The dimension of $V$ is equal to number of columns of $A$ so we know $\operatorname{rank} A + \operatorname{nullity} A = \dim V$ from the Rank-Nullity Theorem for $A$ . ■

From a previous worksheet, we have the following.

Proposition. A linear transformation $T: V \to W$ is an isomorphism if and only if $\mathcal Z(T) = 0$ and $\mathcal R(T) = W$ .

There is another common adjective for isomorphisms.

Definition. A linear transformation $T: V \to W$ is nonsingular if $T$ is an isomorphism. Otherwise, it is called singular.

Computing the null space and range of a linear transformation offers a means to check non-singularity.