We have previously provided two definitions of a vector space:
informal definition: a vector is a finite array of numbers, and a set of such arrays is said to be a vector space if and only if it is closed with respect to taking linear combinations;
formal definition: a vector space is a set equipped with two operations, called vector addition and scalar multiplication, that satisfy a number of axioms.
We have also explained that the simpler informal definition is perfectly compatible with the more formal definition, as a set of numerical arrays satisfies all the properties of a vector space, provided that vector addition and scalar multiplication are defined in the usual manner and that the set is closed with respect to linear combinations.
We now introduce a new concept, that of a coordinate vector, which makes the two definitions almost equivalent: if we are dealing with an abstract vector space, but its dimension is finite and we are able to identify a basis for the space, then we can write each vector as a linear combination of the basis; as a consequence, we can represent the vector as an array, called a coordinate vector, that contains the coefficients of the linear combination. Once we have obtained this simple representation, we can apply the usual rules of matrix algebra to the coordinate vectors, even if we are dealing with an abstract vector space. Not only this is very convenient, but it blurs the differences between the two approaches to defining vectors and vector spaces (at least for the finite-dimensional case).
We are now ready to give a definition of coordinate vector.
Definition Let be a finite-dimensional linear space. Let be a basis for . For any , take the unique set of scalars such thatThen, the vectoris called the coordinate vector of with respect to the basis .
Note that the uniqueness of the scalars is guaranteed by the uniqueness of representations in terms of a basis.
Example Let be a vector space and a basis for it. Suppose that a vector can be written as a linear combination of the basis as follows:Then, the coordinate vector of with respect to is
Example Consider the space of second-order polynomialswhere the coefficients and the argument are scalars. As we have already discussed in the lecture on linear spaces, is a vector space provided that the addition of polynomials and their multiplication by scalars is performed in the usual manner. Consider the polynomialsThese three polynomials form a basis for because they are linearly independent (no combination of them is equal to zero for any ) and they can be linearly combined so as to obtain any of the form above:The coordinate vector of with respect to the basis we have just found is
The addition of two vectors can be carried out by performing the usual operation of vector addition on their respective coordinate vectors.
Proposition Let be a linear space and a basis for . Let . Then, the coordinate vector of with respect to the basis is equal to the sum of the coordinate vectors of and with respect to the same basis, that is,
Suppose that the representations in terms of the basis areso that the coordinate vectors areBy the commutative and distributive properties of vector addition and scalar multiplication in abstract vector spaces, we have thatThus, the coordinate vector of is
The multiplication of a vector by a scalar can be carried out by performing the usual operation of multiplication by a scalar on its coordinate vector.
Proposition Let be a linear space and a basis for . Let and let be a scalar. Then, the coordinate vector of with respect to the basis is equal to the product of and the coordinate vector of , that is,
Suppose the representation in terms of the basis isso that the coordinate vector isBy the associative and distributive properties of scalar multiplication in abstract vector spaces, we have thatThus, the coordinate vector of is
When the elements of a linear space are one-dimensional arrays of numbers (vectors, in the simplest sense of the term), then they coincide with their coordinate vectors with respect to the standard basis. For example, let be the space of all column vectors. Let be its canonical basis, where is a vector whose entries are all , except the -th, which is equal to : Take any Then, is the same as its coordinate vector with respect to the basis , that is,because
Below you can find some exercises with explained solutions.
Let be the vector space of all third-order polynomials. Perform the addition of two polynomialsandby using their coordinate vectors with respect to the basisCheck that the result is the same that you would get by summing the two polynomials directly.
The representations in terms of the basis areThus, the two coordinate vectors areTheir sum isso thatThis is the same result that we obtain by carrying out the addition directly:
Let be the space of all vectors. Consider the basis whereFind the coordinate vector of with respect to the given basis.
We have thatTherefore, the coordinate vector of is
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