# Vector spaces

#### FilipVz

##### New member
Hi,

can somebody help me with the following problem:

Thank you.

MHB Math Helper

#### Deveno

##### Well-known member
MHB Math Scholar
Lazy, hazy proof:

An isomorphism is, among other things, a bijection. So all one needs to do is show 2 things:

1) A linear injection preserves linear independence
2) A linear surjection preserves spanning

These two facts together show that the image under our given isomorphism of a basis for the first vector space is a basis for the second space, and since the isomorphism is bijective, they have the same cardinality.