Respuesta :
By using Gram Schmidt process of orthogonalization, it can be calculated that-
Distance from a vector (2; 3; 1)t to the subspace spanned by the vectors [tex](1; 2; 3)^T , (1; 3; 1)^T[/tex] IS 2.51 units
What is Gram Schmidt process of orthogonalization?
In an inner product space, the set of mathematical operations which converts a set of linearly independent vectors into an orthonormal vector that spans the same set as spanned by the original vectors is called Gram Schmidt process of orthogonalization.
Distance from [tex]b=(2,3,1)^T[/tex] to W= [tex]Span\left \{u_1= (1,2,3)^T,u_2=(1,3,1)^T \right \}[/tex]
Here, Gram Schmidt process of orthogonalization is used
[tex]v_1=u_1=(1,2,3)^T\\ v_2=u_2-\frac{ < u_2,v_1 > }{||v_1||^2}v_1[/tex]
[tex]=(1,3,1)^T-\frac{(1,2,3)^T.(1,3,1)^T}{(1,2,3)^T.(1,2,3)^T}(1,2,3)^T\\ =(1,3,1)^T-\frac{1+6+3}{1+4+9}(1,2,3)^T\\ =(2/7,11/7,-16/14)^T[/tex]
[tex]v_2 =(2/7,11/7,-8/7) ^T[/tex]
[tex]proj_bW=\frac{(2,3,1)(1,2,3)}{(1,2,3)(1,2,3)}(1,2,3)+\frac{(2,3,1)(2/7,11/7,-8/7)}{(2/7,11/7,-8/7)(2/7,11/7,-8/7)}(2/7,11/7,-8/7)[/tex]
=(1.42, 5.11, -0.22)
[tex]b-proj_bW[/tex]=(2,3,1)-(1.42,5.11, -0.22)=(0.58,-2.11,1.22)
Required distance from b onto W=
[tex]||b-pro_bW||=||(0.58,-2.11,1.22)||=\sqrt{(0.58)^2+(-2.11)^2+(1.22)^2}[/tex] = 2.51 units
To know more about Gram Schmidt process of orthogonalization, refer to the link-
https://brainly.com/question/17412861
#SPJ4
