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.