Question

show that two finite dimensional vector space over a given field f are isomorphic if and only if their dimensions are same​

Answer 1

Answer:

Two vector spaces V and W over the same field F are isomorphic if there is a bijection T : V → W which preserves addition and scalar multiplication, that is, for all vectors u and v in V , and all scalars c ∈ F, T(u + v) = T(u) + T(v) and T(cv) = cT(v). The correspondence T is called an isomorphism of vector spaces.

Answer 2

SOLUTION

TO PROVE

Two finite dimensional vector space over a given field F are isomorphic if and only if their dimensions are same

PROOF

Let V and W be isomorphic

Then there exists a linear mapping T : V → W such that T is both one to one and onto

Since T is onto Im T = W

Since T is one to one Ker T = { θ }

Therefore the relation

dim Ker T + dim Im T = dim V gives

dim W = dim V

Conversely let dim V = dim W = n

$\sf{Let \: \: \{ \: \alpha _1, \alpha _2, \alpha _3, .., \alpha _n \: \} \: be \: a \: basis \: of \: V}$

$\sf{Let \: \: \{ \: \beta _1, \beta _2, \beta _3, .., \beta _n \: \} \: be \: a \: basis \: of \: W}$

Then there exists a mapping T : V → W such that

$\sf{T( \alpha _1) = \beta _1, T( \alpha _2) = \beta _2,..,T( \alpha _n) = \beta _n}$

Thus

$\sf{Im T = L \{ \beta _1, \beta _2,.., \beta _ n \}}$

This implies that T is onto

Therefore the relation

dim Ker T + dim Im T = dim V gives

dim Ker T + n = n

⇒ dim Ker T = 0

⇒ Ker T = { θ }

This implies that T is one to one

Since T is both one to one and onto , the vector spaces V and W be isomorphic

Hence the proof follows

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