# To find an orthonormal basis for the row space of $A$.

To find an orthonormal basis for the row space of $$A = \begin{bmatrix} 2 & -1 & -3 \\ -5 & 5 & 3 \\ \end{bmatrix}$$.

Let $$v_1 = (2\ -1 \ -3)$$ and $$v_2 = (-5 \ \ \ 5 \ \ \ 3)$$.

Using the Gram-Schmidt Process, I found an orthonormal basis $$e_1 = \frac{1}{\sqrt{14}} (2\ -1 \ -3)$$ and $$e_2 = \frac{1}{\sqrt{5}} (-1 \ \ \ 2 \ \ \ 0)$$.

So an orthonormal basis for the row space of $$A =\{ e_1,e_2\}$$ .

IS the solution correct?

• Did you try checking if the two vectors you obtained are orthogonal (i.e. their dot product is $0$)? You should also probably show us the steps in your working, so we can see where you went wrong. – Minus One-Twelfth Apr 14 at 2:45
• Even more importantly, have you checked that $v_1$ and $v_2$ are actually elements of the row space? – amd Apr 14 at 3:31

## 1 Answer

Verify your Gram-Schmidt process again.

Note that we have $$V_1=X_1$$ and $$V_2 = X_2-\frac {X_2.V_1}{V_1.V_1}V_1$$

My calculations did not match with yours.