- #1
peripatein
- 880
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Hi,
I am trying to prove that Sp{(a,b),(c,d)} = R^2 if and only if ad-bc≠0.
I am wondering whether the proof below would be considered rigorous.
NB. I am not permitted to make use of number of dimentions.
First direction:
Let Sp{(a,b),(c,d)} = R^2
Hence, (x,y) = alpha(a,b) + beta (c,d)
Hence, x=a*alpha + c*beta; y=b*alpha + d*beta
Hence, beta(ad-bc) = xb-ya
In order that Sp{(a,b),(c,d)} = R^2 and any vector in R^2 could be represented as a linear combination of ((a,b),(c,d)) there has to be a solution for beta, i.e. ad-bc cannot be zero.
Second direction: I am going to try to show that provided that ad-bc≠0, Sp{(a,b),(c,d)}=R^2.
Supposing ad=bc, then (c,d) = (c/a)(a,b) for a≠0, or (c,d) = (d/b)(a,b) for b≠0.
Hence, {(a,b),(c,d)} is linearly dependent over R^2.
Hence, Sp{(a,b),(c,d)} = Sp{(a,b)} whilst a≠0, which is not equal to R^2, OR, Sp{(a,b)} whilst b≠0, which is also not equal to R^2.
Hence, ad-bc≠0 => Sp{(a,b),(c,d)} = R^2
Is the above sufficient? Is the proof correct?
Homework Statement
I am trying to prove that Sp{(a,b),(c,d)} = R^2 if and only if ad-bc≠0.
I am wondering whether the proof below would be considered rigorous.
NB. I am not permitted to make use of number of dimentions.
Homework Equations
The Attempt at a Solution
First direction:
Let Sp{(a,b),(c,d)} = R^2
Hence, (x,y) = alpha(a,b) + beta (c,d)
Hence, x=a*alpha + c*beta; y=b*alpha + d*beta
Hence, beta(ad-bc) = xb-ya
In order that Sp{(a,b),(c,d)} = R^2 and any vector in R^2 could be represented as a linear combination of ((a,b),(c,d)) there has to be a solution for beta, i.e. ad-bc cannot be zero.
Second direction: I am going to try to show that provided that ad-bc≠0, Sp{(a,b),(c,d)}=R^2.
Supposing ad=bc, then (c,d) = (c/a)(a,b) for a≠0, or (c,d) = (d/b)(a,b) for b≠0.
Hence, {(a,b),(c,d)} is linearly dependent over R^2.
Hence, Sp{(a,b),(c,d)} = Sp{(a,b)} whilst a≠0, which is not equal to R^2, OR, Sp{(a,b)} whilst b≠0, which is also not equal to R^2.
Hence, ad-bc≠0 => Sp{(a,b),(c,d)} = R^2
Is the above sufficient? Is the proof correct?