Isomorphic Vector Spaces of R6

[donald1403]

Which vector spaces are isomorphic to R6?

a) M 2,3
b) P6
c) C[0,6]
d) M 6,1
e) P5
f) {(x1,x2,x3,0,x5,x6,x7)}

I know that without showing my work, helper won't answer my question. Since i don't even where to start, all i need is an example. I don't need the complete solution for it. I tried to find examples for isomorphic but I don't quite see which make me understand. So if someone can show step-by-step examples, it would be really helpful.Thx!

 

[matt grime]

The only thing you need to work out is the dimension over R. Or not even that. Which of those are 6 dimensional (real) vector spaces?

As a piece of advice, you might want to remember that P6 is completely meaningless. You might mean what someone else writes as P_6, or P^6, which we can *guess* is the polys of degree <=6 with real coefficients.

 

[donald1403]

yeah i mean P^6. anyway i guess that helps. so u mean if the dimension are same, they can call it isomorphic? do i have to count for one-to-one and onto?
when could be the chance that Transformation is onto but not one-to-one?

 

[Dick]

yeah i mean P^6. anyway i guess that helps. so u mean if the dimension are same, they can call it isomorphic? do i have to count for one-to-one and onto?
when could be the chance that Transformation is onto but not one-to-one?

The question says 'isomorphic as vector spaces'. This just says, as matt says, that they have the same number of basis vectors over the same field. And also, as matt points out, I have no idea what your symbols mean for the vector spaces. I can only guess. So if one of those, perhaps one with a 'C', is supposed to be a six dimensional space but over the complex numbers, then it doesn't count.

 

[matt grime]

yeah i mean P^6.

That still doesn't state what P^6 is. And we have no idea what the {x_1 etc means either. Or C[00,6]). (The continuous functions on the interval [0,6]? Perhaps.

anyway i guess that helps. so u mean if the dimension are same, they can call it isomorphic?

Try to write down an isomorphism between any two vector spaces of the same dimension (over the same field).

do i have to count for one-to-one and onto?

What do you mean?

when could be the chance that Transformation is onto but not one-to-one?

What does it matter? It doesn't matter what linear maps exist between vector spaces that aren't isomorphisms, only whether or not there are any isomorphisms, and that is if and only if they have the same dimension (over the same field).

Exercise: prove this fact. HINT: it was probably proved in your lectures/book.