Does orthogonality depend on inner product?

The notion of inner product allows us to introduce the notion of orthogonality, together with a rich family of properties in linear algebra. Definition. Two vectors u,v ∈ Rn are orthogonal if u · v = 0. Theorem 1 (Pythagorean).

How do you find orthogonal projection in inner product space?

Two vectors are orthogonal if and only if u+v2 = u2 +v2. u + v2 = (u + v) · (u + v) = u · u + u · v + v · u + v · v = u2 + v2 + 2u · v. The theorem follows from the fact that u and v are orthogonal if and only if u · v = 0.

Can orthogonal vectors be linearly dependent?

Orthogonal sets are automatically linearly independent. Theorem Any orthogonal set of vectors is linearly independent.

Are linearly independent vectors orthogonal?

How do you find orthogonality?

To determine if a matrix is orthogonal, we need to multiply the matrix by it’s transpose, and see if we get the identity matrix. Since we get the identity matrix, then we know that is an orthogonal matrix.

What is the condition for the orthogonality of two circle?

1 If two circles intersect in two points, and the radii drawn to the points of intersection meet at right angles, then the circles are orthogonal.

What does it mean when vectors are orthogonal?

Definition. We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero.

Can vectors be linearly independent but not orthogonal?

It is simple to find an example in R2 with the usual inner product: take v=(1,0) and u=(1,1), they are linearly independent but not orthogonal. Indeed, any two vectors in R2 that are not in the same (or opposite) direction, no matter how small the angle between them.

How do you prove two vectors are orthogonal?

We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero.

How do you prove 3 vectors are orthogonal?

Vectors U, V and W are all orthogonal such that the dot product between each of these (UVVWWU) is equal to zero….To construct any othogonal triple we can proceed as follows:

1. choose a first vector v1=(a,b,c)
2. find a second vector orthogonal to v1 that is e.g. v2=(−b,a,0)
3. determine the third by cross product v3=v1×v2.

Can two linearly dependent vectors be orthogonal?

For your true false question, every orthogonal set need not be linearly independent, as orthogonal sets can certainly include the ‘0’ vector, and any set which contains the ‘0’ vector is necessarily linearly dependent.

Does Independent imply orthogonal?

Any pair of vectors that is either uncorrelated or orthogonal must also be independent. vectors to be either uncorrelated or orthogonal. However, an independent pair of vectors still defines a plane. A pair of vectors that is orthogonal does not need to be uncorrelated or vice versa; these are separate properties.

Can two vectors be orthogonal under a different inner product?

Bookmark this question. Show activity on this post. I read that given a vector space V, and two vectors in V, then the two vectors may be orthogonal under one inner product definition but not orthogonal under a different inner product.

What does it mean to specify an alternative inner product?

Specifying an alternative inner product on R n amounts to specifying a positive definite symmetric n × n matrix (whose entries specify the new inner products between the standard basis vectors).

Why is the inner product of X and y equal to zero?

Since the inner product of vectors x and y is equal to zero, the two vectors are orthogonal. For vector z to be orthogonal to both x and y , both inner product calculated above must be equal to zero. Hence the system of equations to solve

What is the inner product of u u and V V?

Then, the inner product of u u and v v is u′v u ′ v. The vectors u u and v v are n ×1 n × 1 matrices where u′ u ′ is a 1×n 1 × n matrix and the inner product u′v u ′ v is a scalar ( 1 ×1 1 × 1 matrix). The inner product is also sometimes called the dot product and written as u ⋅v u ⋅ v.