Hi experts, I need help with this linear algebra problem. Please show me the ans

Question

Hi experts,

I need help with this linear algebra problem. Please show me the answer step by step with clear pictures. Thank you.

and i F Note that B tif, i2, i is an orthogonal set. Also, Let U1 1 v2 let W e the subspace spanned by 101, V2, V3 a. Il point Find the vector in W closest to without inverting any matrices or solving any systems of linear equations. -1 Note that B U1, 2, U3 is an orthogo- and i F b. 1 point Let U1 v2 nal basis for the subspace spanned by B. Find an orthonormal basis for the subspace spanned by B.

Explanation / Answer

(a) Let A be the matrix with v1, v,v3 and the given vector in W, as columns. Then A ==

-1

-1

-1

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1

-3

1

-1

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-7

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We will reduce A to its RREF as under:

Multiply the 1st row by -1; Add 1 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row; Add -1 times the 1st row to the 4th row

Interchange the 2nd row and the 3rd row; Multiply the 2nd row by -1/2

Add 2 times the 2nd row to the 4th row; Multiply the 3rd row by ½

Add 2 times the 3rd row to the 4th row; Multiply the 4th row by ¼

Add 1 times the 4th row to the 3rd row; Add -4 times the 4th row to the 2nd row

Add -1 times the 4th row to the 1st row; Add -1 times the 3rd row to the 1st row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is I4 . This means that the columns of A are linearly independent and hence the given vector in W is not close to any of the vectors v1, v,v3.

(b) B = { v1, v2,v3} is an orthogonal basis for the subspace spanned by B. In order to determine an orthonormal basis for this subspace, we only need to convert v1, v,v3 to unit vectors.

Let u1 = v1/ ||v1|| = (7,0,-4,4)T/(49+0+16+16) = (7,0,-4,4)T/( 81) = 1/9(7,0,-4,4)T = (7/9,0,-4/9,4/9)T.

u2 = v2/ ||v2|| = (4,0,-1,-8)T/(16+0+1+64) = (4,0,-1,-8)T/( 81) = 1/9(4,0,-1,-8)T = (4/9,0,-1/9,-8/9)T. u3 = v3 as v3 is already a unit vector.

Then { u1, u2, u3} is an orthonormal basis for the subspace spanned by B.

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