unit length vector in kernel of matrix

Question

congratulation you all passed festival(new year,christmas),guys i have question related kernel of matrix,namely suppose we have following matrix

$$ A= \begin{bmatrix} 1 & -1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \\ \end{bmatrix} $$

we should find unit-length vector of kernel of matrix,for kernel i think we should find such null space or vector $x$ for which

$A*x=0$ as it is indicated on following wikepedia site

[http://en.wikipedia.org/wiki/Kernel_(matrix)][1]

but i think it has many solution because if we interpret $x$ as $x=[x_1, x_2, x_3]$ we get

$x_1-x_2+0*x_3=0$

$x_3=0$ so we have

$x_1=x_2$

$x_3=0$
how can i continue?please help me

score 4 · Accepted Answer · answered Jan 16 '13 at 18:31

4

Yes, for a vector in the kernel, $x_2$ is arbitrary, $x_1 = x_2$, and $x_3 = 0$, i.e. the vector is $\displaystyle\pmatrix{x_2\cr x_2\cr 0\cr}$. Now use the condition that the length should be $1$.

answered Jan 16 '13 at 18:31

Robert Israel

448,999

1

so it means that $x_2$ equals 1 over square root of 2? – dato datuashvili Jan 16 '13 at 18:33
yes @Nancy R another +1 for helping – dato datuashvili Jan 17 '13 at 05:40

unit length vector in kernel of matrix

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