Editing Coordinate vector (section)

=== Example 1 ===
Let <math>P_3</math> be the space of all the algebraic [[polynomials]] of degree at most 3 (i.e. the highest exponent of ''x'' can be 3). This space is linear and spanned by the following polynomials:
:<math>B_P = \left\{  1,  x,  x^2,  x^3 \right\}</math>

matching
:<math>
    1 := \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} ; \quad
    x := \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} ; \quad
  x^2 := \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} ; \quad
  x^3 := \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}
</math>

then the coordinate vector corresponding to the polynomial
:<math>p \left( x \right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3</math>

is
:<math>\begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix}.</math>

According to that representation, the [[differentiation operator]] ''d''/''dx'' which we shall mark ''D'' will be represented by the following [[matrix (mathematics)|matrix]]:
:<math>Dp(x) = P'(x) ; \quad [D] = 
  \begin{bmatrix}
    0 & 1 & 0 & 0 \\
    0 & 0 & 2 & 0 \\
    0 & 0 & 0 & 3 \\
    0 & 0 & 0 & 0 \\
  \end{bmatrix} 
</math>

Using that method it is easy to explore the properties of the operator, such as: [[invertible matrix|invertibility]], [[hermitian|Hermitian or anti-Hermitian or neither]], spectrum and [[eigenvalues]], and more.