Concept of Basis Vectors and Directional Representation of Vectors

Any given \(N\)-Dimensional Vector \(A\) (where \(N \geq2\)) can be written as a Sum of \(M\) Number of \(N\)-Dimensional Vectors where in each of the Added Vectors is Scaled by a Numerical Value as given in the following

\(A=\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix} = A_{s1} e_1 + A_{s2} e_2 + \cdots + A_{sm}e_m =\begin{bmatrix}e_1 & e_2 & \cdots & e_m\end{bmatrix}\begin{bmatrix}A_{s1}\\A_{s2}\\ \vdots \\A_{sm}\end{bmatrix}\) ...(1)

In equation (1) above, \(A_{1}, A_{2}, ..., A_{n}\) are Components of \(N\)-Dimensional Vector \(A\), \(e_1, e_2, ..., e_m\) are \(M\) Number of \(N\)-Dimensional Vectors and \(A_{s1}, A_{s2}, ..., A_{sm}\) are Numerical Values which Scale the Vectors \(e_1, e_2, ..., e_m\) respectively.

If the \(M\) Vectors \(e_1,e_2, ..., e_m\) are chosen such that they are Linearly Independent, then the Vectors \(e_1, e_2, ..., e_m\) are called the Basis Vectors and the Numerical Values \(A_{s1}, A_{s2}, ..., A_{sm}\) that Scale those Vectors are called the Components of Vector \(A\) corresponding to their respective Basis Vectors.

Since \(M\) Number of \(N\)-Dimensional Vectors can be Linearly Independent only when \(2 \leq M \leq N\), therefore any \(N\)-Dimensional Vector \(A\) can be written as a Scaled Sum of \(2\) to \(N\) Basis Vectors of \(N\)-Dimensions.
For the Numerical Values \(A_{s1}, A_{s2}, ..., A_{sm}\) which Scale the Basis Vectors \(e_1, e_2, ..., e_m\) to have same values as the Actual Components \(A_{1}, A_{2}, ..., A_{n}\) of Vector \(A\) in equation (1) above, following 2 conditions are required
1. Number of Basis Vectors and Numerical Values which Scale the Basis Vectors Must be Same as Number of Acutual Components of the Vector, i.e \(M=N\), so that the equation (1) can be written as
  
  \(A=\begin{bmatrix}e_1 & e_2 & \cdots & e_n\end{bmatrix}\begin{bmatrix}A_{s1}\\A_{s2}\\ \vdots \\A_{sn}\end{bmatrix}=A_{s1} e_1 + A_{s2} e_2 + \cdots + A_{sn}e_n =\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}\) ...(2)
2. The Basis Vector Matrix given by \(\begin{bmatrix}e_1 & e_2 & \cdots & e_n\end{bmatrix}\) in equation (2) above Must be an Identity Matrix (which means that the Basis Vectors are Columns of Identity Matrix), so that the equation (2) can be written as
  
  \(A=\begin{bmatrix}1 & 0& \cdots & 0_n \\ 0 & 1 & \cdots & 0_n \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1_n\end{bmatrix}\begin{bmatrix}A_{s1}\\A_{s2}\\ \vdots \\A_{sn}\end{bmatrix} = A_{s1} \begin{bmatrix}1 \\0 \\ \vdots \\ 0 \end{bmatrix} + A_{s2} \begin{bmatrix}0 \\1 \\ \vdots \\ 0 \end{bmatrix} + \cdots + A_{sn} \begin{bmatrix}0 \\0 \\ \vdots \\ 1\end{bmatrix} =\begin{bmatrix}A_{s1}\\A_{s2}\\ \vdots \\A_{sn}\end{bmatrix} =\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}=A_{1} e_1 + A_{2} e_2 + \cdots + A_{n}e_n \) ...(3)
  
  where \(e_1=\begin{bmatrix}1 \\0 \\ \vdots \\ 0 \end{bmatrix},\hspace{2mm}e_2=\begin{bmatrix}0 \\1 \\ \vdots \\ 0 \end{bmatrix},\hspace{2mm}\cdots,\hspace{2mm}e_n=\begin{bmatrix}0 \\0 \\ \vdots \\ 1 \end{bmatrix}\)
  
  These Basis Vectors \(e_1,e_2, ..., e_n\) forming the Columns of Identity Matrix are also called Identity Orthonormal Basis Vectors or Standard Basis Vectors and have following 2 properties
  1. All Standard Basis Vectors are Unit Vectors that have Numerical Value 1 as one of the Components and have Numerical Value 0 as other Components.
  2. No 2 Standard Basis Vectors have Numerical Value 1 as Same Component.
From equation (3) above we get that Any \(N\)-Dimensional Vector can be written as Sum of \(N\)-Standard Basis Vectors where in each Standard Basis Vector is Scaled by a Component of the Vector.

For example, any \(2\)-Dimensional Vector \(A\) having Components \(A_1\) and \(A_2\) can be given in terms of \(2\)-Dimensional Standard Basis Vectors \(e_1\) and \(e_2\) as follows

\(A= \begin{bmatrix}A_1\\A_2\end{bmatrix} = A_1e_1 + A_2e_2= A_1\begin{bmatrix}1\\0\end{bmatrix} + A_2\begin{bmatrix}0\\1\end{bmatrix}= \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}\begin{bmatrix}A_1\\A_2\end{bmatrix}\) ...(4)

where \(e_1=\begin{bmatrix}1 \\0\end{bmatrix},\hspace{2mm}e_2=\begin{bmatrix}0 \\1 \end{bmatrix}\)

Similarly, any \(3\)-Dimensional Vector \(B\) having Components \(B_1\), \(B_2\) and \(B_3\) can be given in terms of \(3\)-Dimensional Standard Basis Vectors \(e_1\), \(e_2\) and \(e_3\) as follows

\(A= \begin{bmatrix}B_1\\B_2\\B_3\end{bmatrix} = B_1e_1 + B_2e_2 + B_3e_3= B_1\begin{bmatrix}1\\0\\0\end{bmatrix} + B_2\begin{bmatrix}0\\1\\0\end{bmatrix} + B_3\begin{bmatrix}0\\0\\1\end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \end{bmatrix}\begin{bmatrix}B_1\\B_2\\B_3\end{bmatrix}\) ...(5)

where \(e_1=\begin{bmatrix}1 \\0\\0\end{bmatrix},\hspace{2mm}e_2=\begin{bmatrix}0 \\1 \\0 \end{bmatrix},\hspace{2mm}e_3=\begin{bmatrix}0 \\0 \\1 \end{bmatrix}\)

Please note that whenever any Vector is given without Explicitly Specifying it's Basis Vectors (for example when they are given in form of Column Matrix), its automatically assumed that it is given in terms of Standard Basis Vectors. Also All Basis Vectors are themselves Always given in Terms of Standard Basis Vectors.
Basis Vectors can be used as Abstract Units for Directional Representation of Real Vectors in Different Coordinate Systems.

For example, in \(2\)-Dimensional Cartesian Coordinate System,the \(2\)-Dimensional Standard Basis Vectors \(e_1\) and \(e_2\) (as given in equation (4) above) are mapped to Unit Vector in Positive Direction of \(X\)-Axis (represented by \(\mathbf{\hat{i}}\) or \(\mathbf{\hat{x}}\)) and Unit Vector in Positive Direction of \(Y\)-Axis (represented by \(\mathbf{\hat{j}}\) or \(\mathbf{\hat{y}}\)) respectively . Hence, Any Real Vector \(A\) having Components \(A_1\) and \(A_2\) is Directionally Represented in \(2\)-Dimensional Cartesian Coordinate System as

\(A= A_1e_1 + A_2e_2= A_1\mathbf{\hat{i}} + A_2\mathbf{\hat{j}}= A_1\mathbf{\hat{x}} + A_2\mathbf{\hat{y}}\) ...(6)

In equation (6) above, the Components of Real Vector \(A\) given by \(A_1\) and \(A_2\) Directionally Represent a Displacement of \(A_1\) Units along the \(X\)-Axis and a Displacement of \(A_2\) Units along the \(Y\)-Axis.

Similarly, in \(3\)-Dimensional Cartesian Coordinate System,the \(3\)-Dimensional Standard Basis Vectors \(e_1\), \(e_2\) and \(e_3\) (as given in equation (5) above) are mapped to Unit Vector in Positive Direction of \(X\)-Axis (represented by \(\mathbf{\hat{i}}\) or \(\mathbf{\hat{x}}\)), Unit Vector in Positive Direction of \(Y\)-Axis (represented by \(\mathbf{\hat{j}}\) or \(\mathbf{\hat{y}}\)) and Unit Vector in Positive Direction of \(Z\)-Axis (represented by \(\mathbf{\hat{k}}\) or \(\mathbf{\hat{z}}\)) respectively . Hence, Any Real Vector \(B\) having Components \(B_1\), \(B_2\) and \(B_3\) is Directionally Represented in \(3\)-Dimensional Cartesian Coordinate System as

\(B= B_1e_1 + B_2e_2 + B_3e_3= B_1\mathbf{\hat{i}} + B_2\mathbf{\hat{j}} + B_3\mathbf{\hat{k}}= B_1\mathbf{\hat{x}} + B_2\mathbf{\hat{y}} + B_3\mathbf{\hat{z}}\) ...(7)

In equation (7) above, the Components of Real Vector \(B\) given by \(B_1\), \(B_2\) and \(B_3\) Directionally Represent a Displacement of \(B_1\) Units along the \(X\)-Axis, a Displacement of \(B_2\) Units along the \(Y\)-Axis and a Displacement of \(B_3\) Units along the \(Z\)-Axis,
Any Vector given using a particular Set of Basis Vectors can be Ported to different Set of Basis Vectors using the formula for Change of Basis Vectors for a Vector. Such Porting of the Vector does not change the Length/Magnitude/Norm (or Direction in case of Real Vector) of the Vector. It however Changes the Components of the Vector and can Change the Number of Components of the Vector.
Any Set of Basis Vectors can belong to one of the following 6 Types
1. Identity Orthonormal Vectors. Also known as Standard Basis Vectors.
2. Rotated Orthonormal Vectors.
3. Orthogonal Vectors in the Direction of Standard Basis Vectors.
4. Rotated Orthogonal Vectors.
5. Non Orthogonal Vectors.
6. Non Orthogonal Unit Vectors.

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