Question

Do the vectors v1 =   1 2 3   , v2 = ...

Do the vectors v1 =   1 2 3   ,

v2 =   √ 3 √ 3 √ 3   ,

v3   √ 3 √ 5 √ 7   ,

v4 =   1 0 0   form a basis for R 3 ? Why or why not?

(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and a2, where a1 =     1 0 −1 0     , a2 =     0 1 0 −1     . Find a basis for the orthogonal complement V ⊥ in R 4 .

(c) Find a basis for R 4 that includes the vectors a1, and a2. Your answer should be justified.

(d) Explain in words how to generalize your computation in part (c) to obtain a basis for R n that includes a given pair of orthogonal vectors u, v ∈ R n.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3...
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √ 7, v4 = 1 0 0 form a basis for R 3 ? Why or why not? (b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the orthogonal complement V ⊥...
Find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1,...
Find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1, v2 and v3. u = (3, 4, 2, 4) ; v1 = (3, 2, 3, 0), v2 = (-8, 3, 6, 3), v3 = (6, 3, -8, 3) Let (x, y, z, w) denote the orthogonal projection of u onto the given subspace. Then, the components of the target orthogonal projection are
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of vectors in V...
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of vectors in V , and if ⃗v4 ∈ V , then {⃗v1, ⃗v2, ⃗v3, ⃗v4} is also a linear dependent set of vectors in V . 2. Prove that if {⃗v1,⃗v2,...,⃗vr} is a linear dependent set of vectors in V, and if⃗ vr + 1 ,⃗vr+2,...,⃗vn ∈V, then {⃗v1,⃗v2,...,⃗vn} is also a linear dependent set of vectors in V.
If S=(v1,v2,v3,v4) is a linearly independent sequence of vectors in Rn then A) n = 4...
If S=(v1,v2,v3,v4) is a linearly independent sequence of vectors in Rn then A) n = 4 B) The matrix ( v1 v2 v3 v4) has a unique pivot column. C) S is a basis for Span(v1,v2,v3,v4)
If S is the set of vectors in R^4 (S= {v1, v2, v3, v4, v5}) where,...
If S is the set of vectors in R^4 (S= {v1, v2, v3, v4, v5}) where, v1 = (1,2,-1,1), v2 = (-3,0,-4,3), v3 = (2,1,1,-1), v4 = (-3,3,-9,-6), v5 = (3,9,7,-6) Find a subset of S that is a basis for the span(S).
Determine if the vectors v1= (3, 0, -3, 6), v2 = ( 0, 2, 3, 1),...
Determine if the vectors v1= (3, 0, -3, 6), v2 = ( 0, 2, 3, 1), and v3 = (0, -2, 2, 0 ) form a linearly dependent set in R 4. Is it a basis of R4 ?
let v1=[1,0,10], v2=[0,1,0,1] and let W be the subspace of R^4 spanned by v1 and v2....
let v1=[1,0,10], v2=[0,1,0,1] and let W be the subspace of R^4 spanned by v1 and v2. A. convert {v1,v2} into an orhonormal basis of W. Basis = B.find the projection of b=[-1,-2,-2,-1] onto W C.find two linear independent vectors in R^4 perpendicular to W. vectors =
Let H=Span{v1,v2} and K=Span{v3,v4}, where v1,v2,v3,v4 are given below. v1 = [3 2 5], v2 =[4...
Let H=Span{v1,v2} and K=Span{v3,v4}, where v1,v2,v3,v4 are given below. v1 = [3 2 5], v2 =[4 2 6], v3 =[5 -1 1], v4 =[0 -21 -9] Then H and K are subspaces of R3 . In fact, H and K are planes in R3 through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. w = { _______ }
Exercise 6. Consider the following vectors in R3 . v1 = (1, −1, 0) v2 =...
Exercise 6. Consider the following vectors in R3 . v1 = (1, −1, 0) v2 = (3, 2, −1) v3 = (3, 5, −2 )   (a) Verify that the general vector u = (x, y, z) can be written as a linear combination of v1, v2, and v3. (Hint : The coefficients will be expressed as functions of the entries x, y and z of u.) Note : This shows that Span{v1, v2, v3} = R3 . (b) Can R3 be...
Let W be a subspace of R^4 spanned by v1 = [1,1,2,0] and v2 = 2,-1,0,4]....
Let W be a subspace of R^4 spanned by v1 = [1,1,2,0] and v2 = 2,-1,0,4]. Find a basis for W^T = {v is in R^2 : w*v = 0 for w inside of W}