Show that (u1,uzuj) is an orthogonal basis for R3. Then express x as a inear com

Question

Show that (u1,uzuj) is an orthogonal basis for R3. Then express x as a inear combination of the u's. Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace Wof R? Select all that apply A. The vectors must all have a length of 1. B. The distance between any pair of distinct vectors must be constant. C. The vectors must form an orthogonal set. D. The vectors must span W. Which theorem could help prove one of these criteria from another? s. OB. IS (u and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set lfS u, ' .up)isan orthogonal set of nonzero vectors i, R, ten Sis linearly independent and hence is a basis for the subspace spanned by S OD. lfs. up} isa basis in RP, then the members of S span RP and hence form an orthogonal set. Which calculations should be performed next? (Simplily your answers.) Click to select your answerls) Run script "navTorq.%201 70688570,%203,%20

Explanation / Answer

We have u1.u2 = (5,-5,0)T.(2,2,-1)T = 5*2+(-5)*2+0*(-1)= 0, u1.u3 = (5,-5,0)T.(1,1,4)T = 5*1+(-5)*1+0*4= 0 and u2.u3 = (2,2,-1)T.(1,1,4)T = 2*1+2*1+(-1)*4= 0. Hence {u1,u2,u3} is an orthogonal set. Further, let A = [u1,u2,u3,x] =

5

2

1

5

-5

2

1

-2

0

-1

4

1

To show that the set {u1,u2,u3} is a basis for R3 and to determine whether x can be expressed as a linear combinatiion of u1,u2,u3 , we will reduce A to its RREF as under:

Multiply the 1st row by 1/5

Add 5 times the 1st row to the 2nd row

Multiply the 2nd row by ¼

Add 1 times the 2nd row to the 3rd row

Multiply the 3rd row by 2/9

Add -1/2 times the 3rd row to the 2nd row

Add -1/5 times the 3rd row to the 1st row

Add -2/5 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

7/10

0

1

0

5/9

0

0

1

7/18

This implies that the RREF of the matrix [u1,u2,u3] is I3 so that the vectors u1,u2,u3 are linearly independent ( even otherwise, the set { u1,u2,u3} is an orthogonal set, so this set has to be liinearly independent , and span R3 i.e. the set { u1,u2,u3} is a basis for R3. Also, x = (7/10)u1+(5/9)u2+(7/18)u3.

Options C and D, i.e. the vectors must form an orthogonal set and the vectors must span W are necessary for a set of vectors to be an orthogonal basis for a subspace W of Rn.

Option C, i.e. if S = {u1,u2,…,up} is an orthogonal set of non-zero vectors in Rp, then S is linearly independent and hence a basis for the subspace spanned by S.

Option C. The calculations u1.u2 = 0, u1.u3 = 0 and u2.u3 = 0 must be performed next.

5

2

1

5

-5

2

1

-2

0

-1

4

1

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.