Let β and β ′ </msup> be bases for

Question

Let

β

and

β^{'}

be bases for the finite dimensional vector space

V

of dimension n over the field

F

, and let

Q = [I_{V}]_{β^{'}}^{β}

, where

I_{V}

is the identity operator on

V

. I just recently proved that

Q [x]_{β^{'}} = [x]_{β}

for every

x \in V

(twas rather simple), which suggests the title of "basis transformation" matrix or "coordinate transformation" matrix for the matrix

Q

. I am now wondering whether the converse holds.
Let

V^{'}

denote

V

with its elements written with respect to the basis

β^{'}

, and suppose that

Q \in M_{n} (F)

is such that

Q [x]_{β^{'}} = [x]_{β}

for every

x

. Since

L (V^{'} V)

is isomorphic to the matrix algebra

M_{n} (F)

by sending a linear operator to its matrix representation, given

Q

there exists a linear operator

T \in L (V^{'}, V)

such that

Q = [T]_{β^{'}}^{β}

. Hence

[T]_{β^{'}}^{β} [x]_{β^{'}} = [x]_{β}

or

[T (x)]_{β} = [x]_{β}

...
I want to say that this implies

T = I_{V}

, but I can't clearly see what lemma I need in order to make that conclusion.

Answer 1

Your equation

[T (x)]_{β} = [x]_{β}

says that

T (x)

and

x

have the same coordinates in the basis

β

. This means they can be written as the same linear combination of vectors of

β

and thus

T (x) = x

for all

x \in V

, i.e.

T = I_{V}

.

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