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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?

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    $\begingroup$ 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. $\endgroup$ – Minus One-Twelfth Apr 14 at 2:45
  • $\begingroup$ Even more importantly, have you checked that $v_1$ and $v_2$ are actually elements of the row space? $\endgroup$ – amd Apr 14 at 3:31
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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.

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