0
$\begingroup$

How would I use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix? $$A = \begin{bmatrix} 2 & 3 & 1 \\ -1 & -1 & 1 \\ 1 & 0 & 1 \\ 2 & 4 & 1 \end{bmatrix}$$

$\endgroup$

1 Answer 1

3
$\begingroup$

Take first column vector, $v_1 =v_1' = (2, -1, 1, 2)$.Then take second vector, $v_2 = (3, -1, 0,4)$. Then from $v_2$ subtract the component of $v_2$ along $v_1$ and denote it by $v_2'$. i.e. $$v_2' = v_2 - \frac{\langle v_1', v_2\rangle}{\langle v_1', v_1'\rangle} v_1'$$ Then again take $v_3 = (1,1,1,1)$ and subtract it's component of $v_1'$ and $v_2'$ from $v_3$ and denote it by $v_3'$. $$v_3' = v_3 - \frac{\langle v_3, v_1'\rangle}{\langle v_1', v_1'\rangle} v_1-\frac{\langle v_3, v_2'\rangle}{\langle v_2', v_2'\rangle} v_2$$

You get the orthogonal system of vectors $v_1', v_2', v_3'$. Then finally normalize it, you will get set of orthonormal column vector and it is the orthonormal basis of your column space.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.