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Coordinate Vectors

Any vector in a vector space can be represented uniquely by its coordinate vector.

Let V be a vector space with an ordered basis β, and let x be any vector in V:

                  β= x1 xN     xV

We can express x as a linear combination of all the basis vectors in β:

                  x=i=1Naixi   1iN

The coefficients ai in the linear combination above play a pivotal role: they specify how much each basis vector affects the result of the linear combination. Therefore, in recognition of that important role we put them into a column vector denoted by x β:

                  x β= a1 aN

We call the column vector x β the coordinate vector of x relative to the basis β. Each element of the coordinate vector specifies how much contribution is made to the linear combination by the corresponding basis vector in the basis β.

In essence, the coordinate vector tells us which basis vectors make contributions to the linear combination. A zero element in the coordinate vector indicates that the corresponding basis vector does not contribute to the linear combination.

Two examples are provided below in the context of the vector space of polynomials.

Vector Space of Polynomials

Let P3 be the vector space of the polynomials of degree 3, with the ordered basis β:

                  β= 1 x x2 x3

There are four basis vectors in the basis β of P3. All vectors in this space, including the basis vectors themselves, can be uniquely expressed as linear combinations of the four basis vectors.

Example 1

Let fx be a vector in the vector space P3:

                  fx=2+3x+5x2+7x3

The coordinate vector of fx is given by fx β:

                  fx β= 2 3 5 7

In this example, since all the elements of the coordinate vector are non-zero, all basis vectors contribute to the linear combination.

Example 2

Let gx be another vector in the vector space P3:

                  gx=2+7x3

Then the coordinate vector of gx is given by gx β:

                  gx β= 2 0 0 7

In this example, the second and third basis vectors do not make any contribution to the linear combination; only the first and last basis vectors do.

[This content was created with ML, available from the App Store.]