Vectors - Sets of Vectors in n Dimensions

Senin, 10 Februari 2020

Orthogonality. Consider two n-dimensional vectors

x = (x₁, x₂, . . . , x_n),                  y = (y₁, y₂, . . . , y_n).

The inner product of these vectors can be defined,

The vectors are orthogonal if x · y = 0. The norm of a vector is the length of the vector generalized to n dimensions.

     Consider a set of vectors

{x₁, x₂, . . . , x_m}.

If each pair of vectors in the set is orthogonal, then the set is orthogonal.

x_i · x_j = 0     if  i ≠ j

If in addition each vector in the set has norm 1, then the set is orthonormal.

Here δ_ij is known as the Kronecker delta function.

Completeness. A set of n, n-dimensional vectors

{x₁, x₂, . . . , x_n}

is complete if any n-dimensional vector can be written as a linear combination of the vectors in the set. That is, any vector y can be written

Taking the inner product of each side of this equation with x_m,

Thus y has the expansion

If in addition the set is orthonormal, then

Sumber

Labels: Mathematician

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