that's an interesting question. It will depend on what the Vector Space is. For example, R2 over a complex field is NOT a vector space since it isn't closed under scalar multiplication. You can see this by taking a vector in R2, say (1,0) and multiplying it by i. Then i*(1,0) will not be in R2. But if you consider C over the complex field, then it IS a vector space. But it will actually be 1 dimensional since every complex number can be written as a linear combination of 1 complex number. However if you consider C over the real number field, then it is 2 dimensional. The basis could be {1, i}, for instance. Does that make sense?
"I'm curious, how did you write your bold C? Didn't realize you could do that in comments." Its not just a bold, its the symbol for the complex number set. The easiest way is to select and copy the one I typed, save it in a note somewhere, then paste it back when you need it. ℂ ℍ ℕ ℙ ℚ ℝ ℤ Your welcome. ;-)
It does make sense. I struggled awhile wondering whether since ℂ can be written as ℝ(1,0) + ℝ(0,i) then does ℂ2 need to be written as ℝ4 {1(1,0) + 1(0,i) + 2(1,0) + 2(0,i)} or is it just ℝ3 {1(1,0) + 2(1,0) + f(1,2)(0,i)}? It always appears to be the latter if you just look at an illustration on a page, but in reality the former is correct, near as I can tell. That was really my question, hope that makes sense, I am not a mathematician.
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You make it so intuitive and easy to understand! Great job 👏
Thanks!
Can you give an example of what happens if your field is ℂ? Do your dimensions double because ℂ is considered 2 dimensional already?
that's an interesting question. It will depend on what the Vector Space is. For example, R2 over a complex field is NOT a vector space since it isn't closed under scalar multiplication. You can see this by taking a vector in R2, say (1,0) and multiplying it by i. Then i*(1,0) will not be in R2.
But if you consider C over the complex field, then it IS a vector space. But it will actually be 1 dimensional since every complex number can be written as a linear combination of 1 complex number. However if you consider C over the real number field, then it is 2 dimensional. The basis could be {1, i}, for instance.
Does that make sense?
I'm curious, how did you write your bold C? Didn't realize you could do that in comments.
"I'm curious, how did you write your bold C? Didn't realize you could do that in comments."
Its not just a bold, its the symbol for the complex number set. The easiest way is to select and copy the one I typed, save it in a note somewhere, then paste it back when you need it.
ℂ ℍ ℕ ℙ ℚ ℝ ℤ
Your welcome. ;-)
It does make sense. I struggled awhile wondering whether since ℂ can be written as ℝ(1,0) + ℝ(0,i) then does ℂ2 need to be written as ℝ4 {1(1,0) + 1(0,i) + 2(1,0) + 2(0,i)} or is it just ℝ3 {1(1,0) + 2(1,0) + f(1,2)(0,i)}? It always appears to be the latter if you just look at an illustration on a page, but in reality the former is correct, near as I can tell.
That was really my question, hope that makes sense, I am not a mathematician.
The followup question would then be can ℂ2 be mapped to ℍ since those are both dimension ℝ4?