# Thread: Find a basis for...

1. ## Find a basis for...

In R4,

U = <(1,1,0,0) , (0,1,0,2)>.

I want to find a basis for the orthogonal complement U(perp) of U

I know U(perp) can be found by getting the null-space of U

i.e. solving Ux = 0 where x = column vector x1, x2, x3, x4

however im becoming unstuck,

solving Ux = 0 leaves me with (1) x1 + x2 = 0
(2) x2 + 2x4 = 0

2 equations with 3 unknowns means i can solve for 2 variables...?

What variables should i pick as being "free" variables?

2. Originally Posted by piglet
In R4,

U = <(1,1,0,0) , (0,1,0,2)>.

I want to find a basis for the orthogonal complement U(perp) of U

I know U(perp) can be found by getting the null-space of U

i.e. solving Ux = 0 where x = column vector x1, x2, x3, x4

however im becoming unstuck,

solving Ux = 0 leaves me with (1) x1 + x2 = 0
(2) x2 + 2x4 = 0

2 equations with 3 unknowns means i can solve for 2 variables...?

What variables should i pick as being "free" variables?

Strange question... $x_3$ can be anything, and if you fix $x_2\,,\,\,then\,\,\,x_1=-x_2\,,\,\,x_4=-\frac{1}{2}x_2$ , so ...

Tonio

3. Originally Posted by tonio
Strange question... $x_3$ can be anything, and if you fix $x_2\,,\,\,then\,\,\,x_1=-x_2\,,\,\,x_4=-\frac{1}{2}x_2$ , so ...

Tonio

thanks alot, I reckon i can finish the problem from there.