- #1

- 117

- 13

Moved from a technical forum, so homework template missing

**The theorem is as follows:**

All finite dimensional vector spaces of the same dimension are isomorphic

All finite dimensional vector spaces of the same dimension are isomorphic

Attempt:

If T is a linear map defined as :

T : V →W

: dim(V) = dim(W) = x < ∞

& V,W are vector spaces

It would be sufficient to prove T is a bijective linear map:

let W := {w

_{i}}

^{n}

_{i}

like wise let : let V:= {v

_{i}}

^{n}

_{i}

let ω ∈ W & ζ ∈ V

It can be shown:

ω = ∑

_{i}w

_{i}k

_{i}

ζ = ∑

_{i}v

_{i}o

_{i}

The above is a result of the definition of a vector, note k

_{i}and o

_{i}are of an arbitary vector field.

now:

T(ω) = ζ

T(∑

_{i}w

_{i}k

_{i}) = ∑

_{i}v

_{i}o

_{i}

∑

_{i}T(w

_{i}k

_{i}) = ∑

_{i}v

_{i}o

_{i}

thus:

∑

_{i}(T(w

_{i}k

_{i}) - v

_{i}o

_{i}) = 0

therefore:(a bit iffy)

T(w

_{i}k

_{i}) - v

_{i}o

_{i}=0

=>

T(w

_{i}k

_{i}) = v

_{i}o

_{i}

T(ω

_{i})=ζ

_{i}

Hence Surjective

a linear transform is by nature injective, therefore bijective

i have a hunch i am wrong.Any advice :)