### Vector spaces: Coordinates

### The notion of coordinates

In *Coordinate space*, we have seen that, after the selection of a basis, the Euclidean space #\mathbb{E}^3# can be identified with #\mathbb{R}^3#. Here we show that this phenomenon does not stand on its own. In the meantime, we determined that the relevant spaces are in fact vector spaces.

Bases are minimal sets that span a vector space. They have a special property, which we deduce now. If a set of vectors spans a vector space #V#, then any vector of #V# can be written as a linear combination of the set. This expression is not unique in general, but it is if the set is a basis:

Basis criterion

Let #\vec{a}_1,\ldots,\vec{a}_n# be a set of vectors of a vector space #V#, where #n# is a natural number.

The sequence #\vec{a}_1,\ldots,\vec{a}_n# is a basis of #V# if and only if, for each #\vec{x}\in V#, there are unique scalars #x_1,\ldots,x_n# such that

\[

\vec{x}=x_1\cdot \vec{a}_1+\cdots +x_n\cdot \vec{a}_n

\]

We can represent each vector #\vec{x}# from #V# in a unique way by a row of #n# numbers #\rv{x_1,\ldots,x_n}#. This gives rise to the following definition.

Coordinates

Let #n# be a natural number and #\basis{\vec{a}_1, \ldots ,\vec{a}_n}# a basis of a vector space #V#. If

\[

\vec{x}=x_1\cdot\vec{a}_1+\cdots + x_n\cdot\vec{a}_n

\] then the numbers #x_1,\ldots ,x_n# are called the **coordinates** of the vector #\vec{x}# with respect to this basis. The vector #\rv{x_1, \ldots , x_n}# is called the **coordinate vector** of #\vec{x}#. This is itself a vector in #\mathbb{R}^n# if #V# is a real vector space and in #\mathbb{C}^n# if #V# is a complex vector space.

The vector space of all coordinate vectors is called the **coordinate space** of #V#.

Note: the coordinates of a vector depend on the basis used!

The vectors #\rv{ 1 , 1 } # and #\rv{ 1 , -1 } # form a basis of #\mathbb{R}^2#.

The independence can be derived from the fact that #a\cdot \rv{ 1 , 1 } +b\cdot \rv{ 1 , -1 } =\rv{0,0}# implies #a=b=0#. The third *property of the dimension* gives that two independent vectors in the two-dimensional space #\mathbb{R}^2# form a basis of this space.

Determine the coordinates of the vector #\rv{2,-4}# with respect to this new basis.

#\rv{-1,3}#

The coordinates of the vector #\rv{2,-4}# with respect to this new basis are the numbers #c# and #d# for which #\rv{2,-4}=c\cdot\rv{ 1 , 1 } +d\cdot\rv{ 1 , -1 } #. Reading out the coordinates yields:

\[\lineqs{ -c-d+2&=&0\cr -c+d-4 &=&0\cr} \] This system of equations has the unique solution #c=-1# and #d=3#. Therefore, the coordinate vector of #\rv{2,-4}# with respect to the new basis is #\rv{-1,3}#.

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