Can vectors be linearly independent but not orthogonal?

No! Two vectors are linearly dependent if and only if one is a scalar multiple of the other. For example, and are linearly independent, but , so they are not orthogonal.

Is every linearly independent set an orthogonal set?

Not every linearly independent set in Rn is an orthogonal set. If y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.

Is every linearly independent set of vectors in an inner product space is orthogonal?

So right off the bat, no, not all vectors that are linearly independent are orthogonal, because some vector spaces don’t come equipped with inner products with respect to which “orthogonality” can be defined.

Is it true that any orthogonal set of nonzero vectors in an inner product is linearly independent justify your answer with proof?

Since vi is a nonzero vector, its length ‖vi‖ is nonzero. It follows that ci=0. As this computation holds for every i=1,2,…,k, we conclude that c1=c2=⋯=ck=0. Hence the set S is linearly independent.

Why are orthogonal vectors independent?

Orthogonal vectors are linearly independent. If we have n linear independent vectors in Rn, they automatically span the space because the fundamental theorem of linear algebra shows that the image has then dimension n. A vector w ∈ Rn is called orthogonal to a linear space V , if w is orthogonal to every vector v ∈ V .

Why are orthonormal vectors linearly independent?

Theorem 1 An orthonormal set of vectors is linearly independent. where the cj’s(j = 1, 2, …, n) are constants. The set of vectors will be linearly independent if the only constants that satisfy (3) are c1 = c2 = … = cn = 0.

Why are orthogonal vectors linearly independent?

Does orthogonal mean independent?

Simply put, orthogonality means “uncorrelated.” An orthogonal model means that all independent variables in that model are uncorrelated. If one or more independent variables are correlated, then that model is non-orthogonal. The term “orthogonal” usually only applies to classic ANOVA.

Are orthogonal random variables independent or not?

Therefore, orthogonality does not imply independence. See an illustration here. E[XY] is the inner product of the random variables X and Y, defined as the expectation of the product of their pdf’s: ⟨X,Y⟩=E[XY]. where F denotes each random variable’s cumulative distribution function.

What is meant by orthogonal vector?

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. A set of vectors S is orthonormal if every vector in S has magnitude 1 and the set of vectors are mutually orthogonal.