- #1
FunkReverend
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Homework Statement
Find the basis of the solution space W [itex]\subset[/itex] [itex]\Re^{4}[/itex]
of the system of linear equations
[itex]2x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]2x_{3}[/itex] +[itex]3x_{4}[/itex] =0
[itex] _{ }[/itex]
[itex]1x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]3x_{3}[/itex] = 0
Homework Equations
The basis must span W and be independent.
The Attempt at a Solution
Solving the above system, I get
[itex]x_{2}[/itex] = [itex]-x_{1}[/itex] - [itex]x_{3}[/itex]
[itex]x_{4}[/itex] = [itex]\frac{x_{3}-x_{1}}{3}[/itex]
With 2 degrees of freedom, [itex]x_{1}[/itex] and [itex]x_{3}[/itex],
so I must need a 2D basis. I separately fixed [itex]x_{1}[/itex] and [itex]x_{3}[/itex] to 1 and the other to zero and got the following vectors:
[1, -1, 0, -1/3] and [0, -3, 1, 1/3]
I feel like this is right, as I've been looking up some examples, but I'm not sure this spans all the solutions.
Am I on the right track?