Dual basis is a basis of the dual space ↩

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created: 2021-12-13 19:46:21
modified: 2022-01-10 04:13:04

is a vector space over the field and it is finite dimensional. is its dual space.
Statement: The dual basis of is a basis of .

Proof

Suppose is a basis of . Let denote the dual basis.

To show that is a linearly independent list of elements of , suppose are such that

Now for . The equation above thus shows that . Hence is linearly independent.

We also know that , so is a basis of . QED