given a vector space (column vector) of dimension 1 , how i can construct a vector space orthogonal to it?

Question

and what is the dimension of that vector space?

Answer 1

Your space must be embedded in an vector space with inner product. Suppose that your vector space is embedded in n-dimensional real space IR^n, with the dot product as inner product. Let a be a generator of your one-dimensional space (can be any vector in it, except for 0).

Then the orthogonal vector space consists of all vectors x such that a.x = 0, that is

a1*x1 + a2*x2 + ... + an*xn = 0.

This is a (n-1)-dimensional subspace of the original space. You can generate a basis for it as follows:

b1 = (1, 0, 0, 0, ..., 0, -a1/an)
b2 = (0, 1, 0, 0, ..., 0, -a2/an)
...
(If an = 0, choose a different position for this procedure.)

This gives n-1 linearly independent vectors that lie in your subspace. Use the Gramm orthogonalization process to find an orthogonal or orthonormal basis.


Example: suppose you work in IR^4, and your one-dimensional subspace is generated by the vector a = (2, -1, 3, 0). The orthogonal complement is the set
{ (x, y, z, u) | 2x - y + 3z = 0 }

A basis for this 3-dimensional space is
(1, 0, -2/3, 0)
(0, 1, -1/3, 0)
(0, 0, 0, 1)

Answer 2

What you desire to find is the orthogonal complement of your vector space. You can do this by constructing a matrix with rows composed of the vectors in your vector space and then finding the null space of that vector.

Example:

Basis vectors are given by v1 and v2 where:

v1 = { 1 0 0 } ^ Transpose (i.e. write in like a vector)
v2 = {0 1 0} ^ Transpose

(These might have to be orthagonal to eachother; I'm not sure).

Then the matrix formed then v1 and v2 are set up along the rows is given by:

{1 0 0}
{0 1 0}

All you have to do at that point is find the null space of that matrix and you should have your basis vectors for an orthagonal basis.

Also, a theorem that might be useful:

dim(S) + dim(orthagonalSpace(S)) = n

When S is a subspace of R^n

Which should help you out with the dimensionality.

Answer 3

If you have the x-axis as a vector space of dimension 1, the y-axis is orthogonal and remains a vector space of dimension 1

The Cross-Product generates a vector of dimension N which is orthogonal to both vectors of dimension N.

Answer 4

You can't. That is unless you treat this 1-dimensional space as though it were embedded in a larger space. For example, your vector space could have the form: (a,x), where x is any real number and a is fixed. An orthogonal space would be: (y, b) where y is any real number and b is a fixed number.

Answer 5

certainly one of these polynomial has the variety: P(x) = a*x^3 + b*x*2 + c*x + d There are 4 variables. (by using fact the consistent term can substitute is IS a variable.) for that reason, this may be a four-dimensional vector area.