One of the most important concepts in the study of vectors is the concept of a basis. In the English language, the word basis carries the meaning of criterion. Thus, in the sentence “The students were selected on the basis of their results in the MEQ exams” means that the numerical results of some stupid test were used in order to classify the worth of the candidates. Sadly, this type of thing happens a lot and people often disregard the complex characteristics of a person and focus on a single criterion. The meaning of basis in mathematics is more holistic. A basis is a set of criteria that collectively capture all the information about an object.

Let's start with a simple example. If one looks at the HTML code behind the average web-page there will certainly be at least one mention of a colour like background-color:#336699; which should be read as a triplet of values $(33,66,99)$, each one describing how much red, green and blue is needed to create the given colour. The triple $(33,66,99)$ describes the colour “hotmail blue.” This convention for colour representation is called the RGB scale or something I would like to call this the RGB basis. A basis is a set of elements which can be used together to express something more complicated. In our case we have the R, G and B elements which are pure colours and when mixed appropriately they can create any colour. Schematically we can write this as: \[ {\rm RGB\_color}(33,66,99)=33{\mathbf R}+66{\mathbf G}+99{\mathbf B}, \] where we are using the coefficients to determine the strength of each colour component. To create the colour, we combine its components and the $+$ operation symbolizes the mixing of the colours. The reason why we are going into such detail is to illustrate that the coefficients by themselves do not mean much. In fact they do not mean anything unless we know the basis that is being used.

Another colour scheme that is commonly used is the cyan, magenta and yellow (CMY) colour basis. We would get a completely different colour if we were to interpret the same triplet of coordinates $(33,66,99)$ with respect to the CMY basis. To express the “hotmail blue” colour in the CMY basis you would need the following coefficients: \[ {\rm Hotmail Blue} = (33,66,99)_{RGB} = (222,189,156)_{CMY}. \]

A basis is a mapping which converts mathematical objects like the triple $(a,b,c)$ into real world ideas like colours. If there is ever an ambiguity about which basis is being used for a given vector, we can indicate the basis as a subscript after the bracket as we did above.

Look at the bottom left corner of the room you are in. Let's call “the $x$ axis” the edge between the wall that is to your left and the floor. The right wall and the floor meet at the $y$ axis. Finally, the vertical line where the two walls meet will be called the $z$ axis. This is a right-handed $xyz$ coordinate system. It is used by everyone in math and physics. It has three very nice axes. They are nice because they are orthogonal (perpendicular, i.e., at 90$^\circ$ with each other) and orthoginal is good for your life. We will see why that is shortly.

Now take an object of fixed definite length, say the size of your foot. We will call this the unit length. Measure a unit length along the $x$ axis. This is the $\hat{\imath}$ vector. Repeat the same procedure with the $y$ axis and you will have the $\hat{\jmath}$ vector. Using these two vectors and the property of addition, we can build new vectors. For example, I can describe a vector pointing at 45$^\circ$ with both the $x$ axis and the $y$ axis by the following expression: \[ \vec{v}=1\:\hat{\imath}+ 1\:\hat{\jmath}, \] which means measure one step out on the $x$ axis, one step out on the $y$ axis. Using our two basis vectors we can express any vector in the plane of the floor by a linear combination like \[ \vec{v}_{\mathrm{spoint\ on\ the\ floor}}=a\:\hat{\imath}+b\:\hat{\jmath}. \] The precise mathematical statement that describtes this situation is that the basis formed by the pair $\hat{\imath}$,$\hat{\jmath}$ span the two dimensional space of the floor. We can extend this idea to three dimensions by specifying the coordinates of any point in room as a weighted sum of the three basis vectors: \[ \vec{v}_{\mathrm{point\ in\ the\ room}}=a\:\hat{\imath}+b\:\hat{\jmath}+c\:\hat{k}, \] where $\hat{k}$ is the unit length vector along the $z$ axis.

In the case where it is clear which coordinate system we are using in a particular situation, we can take the liberty to omit the explicit mention of the basis vectors and simply write $(a,b,c)$ as an ordered triplet which contains only the coefficients. When there is more than one basis in some context (like in problems where you have to change basis, then for every tuple of numbers we should be explicit about which basis it refers to. We can do this by putting a subscript after the tuple. For example, the vector $\vec{v}=a\:\hat{\imath} + b\:\hat{\jmath}+c\:\hat{k}$ in the standard basis is referred to as $(a,b,c)_{\hat{\imath}\hat{\jmath}\hat{k}}$.

It is hard to over-emphasize the importance of the notion of a basis. Every time you solve a problem with vectors, you need to be consistent in your choice of basis, because all the numbers and variables in your equations will depend on it. The basis is the bridge between real world vector quantities and their mathematical representation in terms of components.