39. Kernels

39. Kernels

Definition of the kernel

Recall the following definitions:

Definition:

f is a linear map if

  • f(v+w)=f(v)+f(w) for all v,w

  • f(λv)=λf(v) for all v and all λ𝐑 .

Definition:

V is a linear subspace if

  • v,wV implies v+wV

  • vV implies λvV for all λ𝐑 .

These two definitions are very similar. We will exploit this in the next two videos: given a linear map f:𝐑n𝐑m , we will associate to it two subspaces ker(f)𝐑n (the kernel of f ) and im(f) (the image of f ).

Definition:

The kernel of f is the set of v𝐑n such that f(v)=0 . (If m=n then this is just the 0-eigenspace of f ).

Example:

Let f:𝐑3𝐑3 be the map f(v)=Av for A=(100010000) . Note that f(xyz)=(xy0) . This is the vertical projection to the xy -plane.

Vertical projection to xy-plane

The kernel of f is the z -axis (blue in the figure; these are the points which project vertically down to the origin). That is ker(f)={(00z):z𝐑} .

Example:

Recall the example A=(10-101-1) (going from 𝐑3 to 𝐑2 ) from Video 3. This projects vectors into the plane; if we think of 𝐑2 as the xy -plane then we can visualise this map as the projection of vectors in the (-1-1-1) -direction until they live in the xy -plane.

Projection along (-1,-1,-1) to xy-plane

We described this as projecting light rays in the (-1-1-1) direction. In this case, the kernel of A is precisely the light ray which hits the origin, which is the line {(xxx):x𝐑} (light blue in the picture).

Kernel is a subspace

Lemma:

The kernel is a subspace.

Given v,wker(f) , we need to show that v+wker(f) . Since v,wker(f) , we know that f(v)=f(w)=0 . Therefore f(v+w)=f(v)+f(w) (since f is linear) =0+0=0 , so v+wker(f) . Similarly, f(λv)=λf(v)=λ0=0 .

Remarks

  • 0ker(f) for any linear map f because f(0)=0 .

  • If f is invertible then ker(f)={0} : if vker(f) then v=f-1(0)=0 .

  • The "kernel" in a nut is the little bit in the middle that's left when you strip away the husk. If f(v)=Av then we can think of ker(f) as the space of solutions to the simultaneous equations Av=0 , which is the intersection of the hyperplanes A11v1++A1nvn=0,,Am1v1++Amnvn=0.

    In other words, it's the little bit left over when you've intersected all these hyperplanes.

Simultaneous equations revisited

Lemma:

Consider the simultaneous equations Av=b (A is an m -by-n matrix and b𝐑m ). Let f(v)=Av . The space of solutions to Av=b , if nonempty, is an affine translate of ker(f) .

Example:

If A=(100010000) (so f is vertical projection to the xy -plane) then Av=b has a solution only if b is in the xy -plane, and in that case it has a whole vertical line of solutions sitting above b .

Vertical projection to xy-plane again

This vertical line of solutions is parallel to the kernel of f (the z -axis), i.e. it is a translate of the kernel.

We saw this lemma earlier in a different guise in Video 19. Namely, we saw that if v0 is a solution to Av=b then the set of all solutions is the affine subspace v0+U where U is the space of solutions to Av=0 . In other words, U=ker(f) .

In particular, we see that if Av=b has a solution then it has a k -dimensional space of solutions, where k is the dimension of ker(f) .

Remember that the space of solutions has dimension equal to the number of free variables when we put A into reduced echelon form. For example, A=(100010000) is in reduced echelon form with two leading entries and one free variable, which is why we get 1-dimensional solution spaces.

Definition:

The nullity of A (or of f ) is the dimension of ker(f) (i.e. the number of free variables of A when put into reduced echelon form).

Our goal for the next video is to prove the rank-nullity theorem which gives us a nice formula relating the nullity to another important number called the rank.