Problem #9: Find a basis for the orthogonal complement of the subspace of R4 spanned by the following vectors.V₁ = (1, -1, 1, 6), V2 = (2,-1, 8, 4), v3 = (-1, 2, 5, -14)The required basis can be written in the form{(x, y, 1, 0), (z, w, 0, 1)}.Enter the values of x, y, z, and w (in that order) into the answer box below, separated with commas.Problem #9:

The given vectors .The aim is to find the basis for the orthogonal complement of the subspace of…

Answered: Find a basis for the orthogonal…

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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There are four vectors (1,a,b,c), (1,1,1,1), (1,2,3,1), (3,4,5,3). If the vector space (subspace) consisting of the linear combination of these vectors is T, find the condition that the number of vectors constituting the basis of T will be 3.
Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 0, 1), v3 = (5, 0, 1), v4 = (0, 0, -2) V2 and v4 form a basis for span {v1, V2, V3, V4}. V1 and v4 form a basis for span {v1, V2, V3, V4}. V1 and v3 form a basis for span {v1, V2, V3, V4}. Ov1 and v2 form a basis for span {v1, V2, V3, V4}. V2 and v3 form a basis for span {v1, V2, V3, V4}. V3 and v4 form a basis for span {v1, V2, V3, V4}.
Find a basis for the orthogonal complement of the subspace of Rª spanned by the following vectors. V1 = (1,-1,7,3), V2 = (2,-1,5,2), V3 = (1, 0, -2, -1) The required basis can be written in the form {(x, y, 1, 0), (z, w, 0, 1)}.
Find a basis of the subspace of R' defined by the equation -5x1- 5x2+ 2x3 = 0. Basis:
The set of vectors {(1,1,0,0),(0,-1,0,0),(0,0,1,1),(1,0,1,1)} is linearly dependent. What is a valid expression of the last vector as a linear combination of the others? a. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)-(0,0,1,1) b. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) c. (1,0,1,1)=-(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) d. (1,0,1,1)=2(1,1,0,0)+2(0,-1,0,0)+2(0,0,1,1)
Consider the subspace S of the Euclidean inner product space R+ 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*((-.-3.1).(1.2.-4,-3).(4,2,1,1)) OB. © ³ {(1,1,2,4), ( − 1, − 1,0,2 A. -1,0,2), (12. 2.-3,1)} O C. None in the given list. OD. {(1,1,1,1),(1,1,2,4), (1,2,-4,-3) } OE {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) }
I know the basis converted to orthogonal is {<2, 1, -1, 1>,<11, -12, 5, -5>,<-1, -3, 0, 5>} and that the magnitude of these will be sqrt(7), sqrt(315), sqrt(35) - but I'm not sure how to use these to convert the basis to orthonormal.
Show that the vectors u₁ = (3, 1,-4), = (3,1, –4), u₂ = (2,5,6), and u} = = (1, 4, 8) form a basis in R³. Find the coordinates of (1, 1, 1) in this basis.
3) Find a basis of F3 containing the vectors (1, 2, -3), (2, 6, -3)
Find a basis for the subspace of R° that is spanned by the vectors V1 = (1,0, 0), v2 = (1, 0, 1), v3 = (2, 0, 1), v4 = (0, 0, -2) Vị and v2 form a basis for span {V1, V2, V3, V4}. V1 and v3 form a basis for span {v1, V2, V3, V4}. V2 and v4 form a basis for span {v1, V2, V3, V43. V1 and v4 form a basis for span {v1, V2, V3, V4}. V2 and v3 form a basis for span {v1, V2, V3, V43. V3 and v4 form a basis for span {v1, V2, V3, V4}. O All of the above are correct.
Suppose V is a subspace of R" and suppose {v1, v2, v3} is a basis of V. Decide if the following sets of vectors are a basis for V: (i) {v2, v1 – 503, 2v3} (ii) {v2, v1 – 503, 203, 302 + 703 – v1} (iii) {202 – v3, v1}

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