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Consider the subspace S of the Euclidean inner product space R4 spanned by the vectors v₁ = (1,1,1,1), v₂= (1,1,2,4), v₂=(1,2,-4,-3). Find an orthogonal basis of S. O A {(1,1,1,1),(1,1,2,4), (1,2,-4,-3)} B{(1,1,2,4),(-1,-1.0.2) (₁.-3.1)} OC. {(2,-3,1),(1,2,-4,-3), (4,2,1,1)} OD. None in the given list. OE. {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2)}

Consider the subspace S of the Euclidean inner product space R4 spanned by the vectors v₁ = (1,1,1,1), v₂= (1,1,2,4), v₂=(1,2,-4,-3). Find an orthogonal basis of S. O A {(1,1,1,1),(1,1,2,4), (1,2,-4,-3)} B{(1,1,2,4),(-1,-1.0.2) (₁.-3.1)} OC. {(2,-3,1),(1,2,-4,-3), (4,2,1,1)} OD. None in the given list. OE. {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2)}

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the subspace S of the Euclidean inner product space R4 spanned by the vectors v₁=(1,1,1,1), v (1,1,1,1), an orthogonal basis of S. OA. {(1,1,1,1),(1,1,2,4),(1,2,-4,-3)} {(1.1.2.4),(-1,-1,0.2).(-3.1)} {(-3.1). (1.2.-4.-3), (4,2,1,1)} D. None in the given list. E. {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) } v₂=(1,1,2,4), v,= (1,2.–4, − 3) . Find 3
Transcribed Image Text:Consider the subspace S of the Euclidean inner product space R4 spanned by the vectors v₁=(1,1,1,1), v (1,1,1,1), an orthogonal basis of S. OA. {(1,1,1,1),(1,1,2,4),(1,2,-4,-3)} {(1.1.2.4),(-1,-1,0.2).(-3.1)} {(-3.1). (1.2.-4.-3), (4,2,1,1)} D. None in the given list. E. {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) } v₂=(1,1,2,4), v,= (1,2.–4, − 3) . Find 3
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Step 1

The given subspace S of the Euclidean inner product space  ℝ4 spanned by the vectors ,

v1=1,1,1,1 , v2=1,1,2,4 , v3=1,2 ,-4,-3

(.) Othogonal basis :                                                                   A basis of a subspace S of   of an inner product space is said to  be   orthogonal basis if  ,                                        (u,v) = 0  for all  u,v∈S such that u≠v   i.e  inner product of every two distinct   vectors of   S  is zero .

 where  (u,v) denotes an inner product of   the vectors  u and v .

(.) Inner product :                                                                     Inner product of two vectors  u=(u1,u2,u3,u4) and  v=v1,v2,v3,v4 is given by ,

     (u,v) = u1v1+u2v2+u3v3+u4v4

(.)  A subspace has infinitely many basis . And every basis has same number of vectors .

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