math

Find an orthonormal basis for the subspace of R^3 consisting of all vectors(a, b, c) such that a+b+c = 0.

The subspace is two-dimensional, so you can solve the problem by finding one vector that satisfies the equation and then by constructing another solution which is orthogonal to it.

E.g. if you take the first basis to be (1,-1,0) up to normalization, then the second basis vector (a,b,c) must have inner product zero with the first basis vector. Therefore:

a - b = 0

We can take, e.g. a = 1 and b = 1.

a + b + c = 0, therefore c = -2

You only have to normalize these two vectors.

In general linear algebra problems you should avoid this way of solving problems, i.e. solving equations to find orthonormal vectors. In this case it was easy, because we only had to find two vectors.

So, let's do this problem again using the sandard way

Let's start by writing down two arbitrary independent vectors, e.g.:

e1 = (1,-1,0)

e2 = (1,0,-1).

Then you use the Gram-Schmidt method: Take the first normalized basis vector to be:

f1 = e1/|e1| = 1/sqrt[2] (1,-1,0)

Subtract from e2 the projection of e2 on f1 to obtain a vector that is orthogonal to f1:

e2' = e2 - <e2,f1>f1.

The fact that e2' is orthogonal to f1 is easy to check:

<e2',f1> = <(e2 - <e2,f1>f1),f1> =

<e2,f1 > - <e2,f1><f1,f1>

Because f1 is normalized to 1,
<f1,f1> (which always equals the square of the norm) equals 1 and we see that <e2',f1> = 0

We can thus take the second orthonormal basis vector to be:

f2 = e2'/|e2'|

e2' = e2 - <e2,f1>f1 =

(1,0,-1) - 1/2 (1,-1,0) =

(1/2,1/2,-1)

f2 = sqrt[1/6] (1,1,-2)

So, no equation solving is necessary to find the orthonormal set from a set of linear independent vectors.

If you had to orthonormalize a larger set of independent vectors, then the next step would have been:

e3' = e3 - <e3,f1>f1 - <e3,f2>f2

f3 = e3'/|e3'|

e4'= e3 - <e3,f1>f1 - <e3,f2>f2-<e3,f3>f3

f4 = e4'/|e4'|

etc.

  1. 👍
  2. 👎
  3. 👁

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    if vectors a+2b and 5a-4b are perpendicular to each other and a and b are unit vectors. Find the angle between a and b.

  2. linear algebra

    what is the basis of a subspace or R3 defined by the equation 2x1+3x2+x3=0 I know the basis is vectors and but how do you get that?

  3. Calculus

    Sketch the vector field F⃗ (r⃗ )=2r⃗ in the plane, where r⃗ =⟨x,y⟩. Select all that apply. A.All the vectors point away from the origin. B. The vectors increase in length as you move away from the origin. C. All the

  4. mathematics

    given vectors u=-9i+8j and v=7i+5j find 2u-6v in terms of unit vectors i and j

  1. linear

    Let W be the subspace of R5 spanned by the vectors w1, w2, w3, w4, w5, where w1 = 2 −1 1 2 0 , w2 = 1 2 0 1 −2 , w3 = 4 3 1 4 −4 , w4 = 3 1 2 −1 1 , w5 = 2 −1 2 −2 3 . Find a basis for W ⊥.

  2. Calculus

    Given vectors u= (1,-1,3) and v= (-4,5,3) find an ordered triple that represents vectors 3u-2v. a.(16,1,-7) b.(11,-13,3) c.(10,-12,1) d.(7,-7,16) I don't want you to straight up give me the answer. I need someone to teach me how

  3. Math

    Find all vectors v in 2 dimensions having ||v ||=5 where the i -component of v is 3i vectors:

  4. physics-vectors

    If the magnitude of the sum of two vectors is greater than the magnitude of either vector, then: If the magnitude of the sum of two vectors is less than the magnitude of either vector, then: a. the scaler product of the vectors

  1. physics

    A football player runs the pattern given in the drawing by the three displacement vectors , , and . The magnitudes of these vectors are A = 4.00 m, B = 15.0 m, and C = 18.0 m. Using the component method, find the (a) magnitude and

  2. math

    given that vectors(p+2q) and (5p-4q) are orthogonal,if vectors p and q are the unit vectors,find the dot product of vectors p and q?

  3. Calculus/Vectors

    Calculate the area of the parallelogram with sides consisting of the vectors: a = (1,2,-2) b = (-1,3,0)

  4. PreCalculus

    Given vectors u=7i + 3j and v = -4i + 3j, find 5u-4v in terms of unit vectors. I am just beginning the study of vectors and have this problem. Could someone help? Thanks.

You can view more similar questions or ask a new question.