Question

A is a 3x4 matrix where A= (1 1 0 0 \ -1 3 0 1 \ -3 1 -2 1)
Find an orthonormal basis for the row space of the matrix using the Gram-Schmidt Process

Answer 1

Let the row space of the matrix $A$ is denoted by $\{ u_1,u_2,u_3 \}$ .

Where $u_1=(1,1,0,0),u_2=(-1,3,0,1),u_3=(-3,1,-2,1)$

Let the orthonormal basis of the row space of $A$ is $\{w_1,w_2,w_3 \}$ .

Now, applying Gram-Schmidt Orthogonalization process ,

$w_1=u_1=(1,1,0,0)$

$$w_2=u_2-\frac{<u{_2},w{_1}>}{<w_1,w_1>}w_1$$

$=u_2-\frac{2}{2}w_1=(-2,2,0,1)$

$$w_3=u_3-\frac{<u_3,w_1>}{<w_1,w_1>}w_1-\frac{<u_3,w_2>}{<w_2,w_2>}w_2$$

$=u_3-\frac{-2}{2} w_1- \frac{9}{9} w_2$

$=(-3,1,-2,1)+(1,1,0,0)-(-2,2,0,1)$

$=(0,0,-2,0)$