- #1

- 751

- 78

## Summary:

- Exercise on linear transformations and dimensional theorem.

## Main Question or Discussion Point

Let ##V## be a real vectorspace of finite dimension ##n##. Let ##L, K:V \rightarrow \Re## be linear transformations so that ##ker(L) \subset ker(K)##. Then there's a parameter ##\lambda \in \Re## so that ##K=\lambda L##

a) Show that ##K=\lambda L## holds when ##K=0##.

b) Suppose that ##K \neq 0##. Compute ##dim(kerK)## and show that ##dim(kerK)=dim(kerL)##.

c) Show that ##K=\lambda L## holds when ##K \neq 0##.

a) Here I think we just have to set ##\lambda=0##and then the equation holds. However I am not that convinced (it seems too easy).

b) To compute ##dim(kerK)## I used the dimension theorem for the linear transformation ##K:V \rightarrow \Re##

$$dim(V) = dim(kerK) + dim(Imk)$$

We are given that ##dim(V)=n## and the co-domain of both linear transformations is ##\Re##. Thus I get

$$dim(kerK)=n-1$$

To compute ##dim(kerL)## I used the dimension theorem for the linear transformation ##L:V \rightarrow \Re##

$$dim(V) = dim(kerL) + dim(ImL)$$

We are given that ##dim(V)=n## and the co-domain of both linear transformations is ##\Re##. Thus I get

$$dim(kerL)=n-1$$

Then indeed we get $$dim(kerL)=dim(kerK)$$ Is this OK?

c) I do not really know how to approach this section. Could you please give a hint?

Any help is appreciated.

Thanks.

a) Show that ##K=\lambda L## holds when ##K=0##.

b) Suppose that ##K \neq 0##. Compute ##dim(kerK)## and show that ##dim(kerK)=dim(kerL)##.

c) Show that ##K=\lambda L## holds when ##K \neq 0##.

a) Here I think we just have to set ##\lambda=0##and then the equation holds. However I am not that convinced (it seems too easy).

b) To compute ##dim(kerK)## I used the dimension theorem for the linear transformation ##K:V \rightarrow \Re##

$$dim(V) = dim(kerK) + dim(Imk)$$

We are given that ##dim(V)=n## and the co-domain of both linear transformations is ##\Re##. Thus I get

$$dim(kerK)=n-1$$

To compute ##dim(kerL)## I used the dimension theorem for the linear transformation ##L:V \rightarrow \Re##

$$dim(V) = dim(kerL) + dim(ImL)$$

We are given that ##dim(V)=n## and the co-domain of both linear transformations is ##\Re##. Thus I get

$$dim(kerL)=n-1$$

Then indeed we get $$dim(kerL)=dim(kerK)$$ Is this OK?

c) I do not really know how to approach this section. Could you please give a hint?

Any help is appreciated.

Thanks.