We give a brief summary of the vector space theory covered today.

Next, we recap the key ideas in the proof of the dimension formula:

.

If is a basis for , and are bases for and resp., then is a basis for .

To show the linear independence, we consider

From this we deduce

Keypoints:

- The LHS is in while the RHS is .

Thus . - As is a basis for ,

.

Rearranging, we get - As is a basis for , we infer .
- Put into , .
- As is a basis for , we infer .

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