Vector calculus in plane and space: Bases, Coordinates and Equations
The notion of basis
In order to carry out actual calculations it is useful to describe vectors using numbers. The concepts we require for this are basis and coordinate.
Basis of plane and space
- A basis (plural: bases) of the plane consists of two vectors and , which are not on the same line through . Each vector of the plane can now be written in its own unique way as a linear combination of the vectors , :
for certain numbers and unique to . The numbers , are called the coordinates of the vector relative to the basis. - A basis of the space consists of three vectors , , and , which are not in the same plane as . Each vector can be written as a linear combination of these three vectors:
where the scalars , , and are uniquely determined. The vectors , , and form a basis of the space, and the numbers , , and , are called the coordinates of the vector with respect to the basis. - The vectors and form the basis of , which we call the standard basis of .
- The vectors , , and , form the basis of , which we call the standard basis of .
The coordinates in the plane and in the space , which have been previously defined, are the coordinates as defined here for the standard basis:
for and similarly, but with an additional term , for in .
We will now check why the coordinates are uniquely determined by a basis of the plane: say . What follows, after subtracting on both sides:
Because and are not on the same line as , we must conclude with the Criteria for two vectors on a line through the origin that and , in other words and . This shows that the coordinates and are unique.
The same goes for the space: say
This can be rewritten as
If , then after multiplying both sides with the the scalar , we see that is a linear combination of and ; this means that is in the plane through , , and , a contradiction to the assumption. What should apply is: . With the same reasoning for the indices and , rather than , we deduce that and . This shows that the coordinates , , and are unique.
We next discuss how to recognize a basis.
Two characteristics of a basis
We fix an origin of the plane and the space.
- Two vectors in the plane, respectively three vectors in the space, form a basis if and only if each vector of the plane, respectively the space, is a linear combination of these.
- Two vectors in the plane, respectively three vectors in the space, form a basis if and only if the only way to write the origin as a linear combination of these vectors, is when all scalars are equal to .
No
To determine this, we use the second characteristic of a basis. Suppose there are two scalars and , so . and satisfy the system of linear equations
This shows that can be written as the linear combination
The first characteristic of a basis can also be used to derive this. If is a linear combination of and , with scalars and , those for the scalars satisfy the system of linear equations
To determine this, we use the second characteristic of a basis. Suppose there are two scalars and , so . and satisfy the system of linear equations
whose solutions are: . In particular, is a solution.
This shows that can be written as the linear combination
wherein are scalars that are not both equal to . The second characteristic of a basis gives that and do not form a basis.
The first characteristic of a basis can also be used to derive this. If is a linear combination of and , with scalars and , those for the scalars satisfy the system of linear equations
This shows that the vector cannot be written as a linear combination of and . The first characteristic of a basis gives that and do not form a basis.
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