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Finding a basis for a subspace

Problem Statement
Find a basis for the subspace in R[SUP]4[/SUP] consisting of the form (a,b,c,d) when c=2a+2b and d=a-5b
Relevant Equations
c=2a+2b and d=a-5b
I had assumed that we had to put our values into a matrix so I did [1 2 -1 0; 1 -5 0 -1] and then I would do a=[1; 1] and repeat for b, c, and d. This is incorrect however. I also thought that it could be {(1, 2, -1, 0),(1, -5, 0, -1)} however this was not the answer, and I am unsure of what do to next and hints would be appreciated.

Thank you.


Science Advisor
Homework Helper
Insights Author
Gold Member
I guess I would try this:$$
(a,b,c,d) = a(1,0,?,?) + b(0,1,?,?) $$
and fill in the ? spaces to make it work.


Science Advisor
Gold Member
One way that helps me make sense is to first determine the number of free variables in your subspace, i.e., variables that can take on any value. Then you choose to assign values by convenience to these (usually value 1, 0 are used/convenient).

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