Isomorphic Vector Spaces of R6

In summary, the conversation discusses the topic of isomorphic vector spaces and how to determine if a given vector space is isomorphic to R6. The key factor in determining isomorphism is the dimension of the vector spaces, which should be the same over the same field. The symbols used in the conversation, such as P6 and C[0,6], have unclear meanings and may not be valid vector spaces. The conversation also mentions the importance of finding an isomorphism between two vector spaces and notes that it is not necessary to consider one-to-one and onto transformations for non-isomorphic vector spaces.
  • #1
donald1403
16
0
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!
 
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  • #2
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.
 
  • #3
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?
 
  • #4
donald1403 said:
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.
 
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  • #5
donald1403 said:
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.
 

What is an isomorphic vector space?

An isomorphic vector space is a mathematical concept in linear algebra where two vector spaces have the same structure and can be mapped onto each other in a one-to-one correspondence. This means that they have the same number of dimensions and operations can be carried out in the same way in both spaces.

What is the importance of isomorphic vector spaces?

Isomorphic vector spaces allow for the study and comparison of different vector spaces. They also make it easier to generalize concepts and theorems across different spaces.

How do you know if two vector spaces are isomorphic?

Two vector spaces are isomorphic if there exists a bijective linear transformation between them. This means that the transformation is both one-to-one and onto, and preserves the operations and structure of the vector spaces.

What is the dimension of an isomorphic vector space?

The dimension of an isomorphic vector space is the same as the dimension of the original vector space. This is because isomorphic vector spaces have the same number of dimensions and structure.

What are some examples of isomorphic vector spaces of R6?

Examples of isomorphic vector spaces of R6 include R6 itself, as well as any other vector space with 6 dimensions, such as the space of 6-dimensional polynomials or the space of 6x6 matrices.

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