Image and kernel

Overview §

The image and kernel are two subspaces associated with a linear map.

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Images and kernels of linear maps §

Let $A:V\to W$ be linear.

Definition: Image

The image of $A$ is what $A$ the set of all vectors “traced out” by plugging vectors into the map.

$$\text{im}(A)=\{Av\mid v\in V\}=\{w\in W\mid {\exists }_{v\in V}Av=w\}\subset W$$

We can compare this definition of the image with its ”set-theoretic counterpart”: the image is the set of all vectors produced by the linear function.

Definition: Kernel

The kernel of $A$ is the set of all vectors sent to zero by $A$.

$$\text{ker}(A)=A^{-1}(\vec{0})=\{v\in V\mid Av=\vec{0}\}\subset V$$

In other words, the kernel is the set of vectors for which the information carried by $v\in V$ is lost when passed to $W$.

The image and kernel have the following properties:

• (Lemma 46) If $A:V\to W$ is linear, then $\text{ker}(A)\subset V$ and $\text{im}(A)\subset W$ are indeed subspaces.
• (Lemma 47) If $A:V\to W$ is linear, then $A$ is injective if and only if $\text{ker}(A)=\{\vec{0}\}$.
  • Analogous: $A\text{ is surjective}⟺A=W$.
• (Lemma 48) If $A:V\to W$ is injective and $\{v_1,\cdots ,v_n\}\subset V$ is linearly independent, then $\{A{v}_1,\cdots ,A{v}_n\}\subset W$ is linearly independent.

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Examples of image and kernel §

#wip

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Rank and nullity §

Definition: Rank and nullity

Let $A:V\to W$ be linear.

• The rank of $A$ is the dimension of its image; $\text{rank}(A)=\text{dim im}(A)$.
• The nullity of $A$ = dimension of its kernel; $\text{null}(A)=\text{dim ker}(A).$

• (Theorem) Rank-nullity