Answer to Question #106513 in Linear Algebra for raymond

Question #106513
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
1
Expert's answer
2020-03-26T13:41:35-0400

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)"


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