4.2 Standard basis vectors of vector spaces

What are standard basis for vector spaces

• the standard basis of the polynomial vector space $P_k$ is $(1,x,x^2,\dots ,x^k)$

  • much like1.5 Basis, but instead of having linearly independent basis, we have separate polynomials

    • ex: standard basis of $P_3$ is $(1,x,x^2,x^3)$ dim($P_3$)=4

  • $a+bx+c{x}^2+d{x}^3\in P_3$ can be written as a linear combination of the standard basis vectors:
    • $a(1)+b(x),c(x^2)+d(x^3)$

Standard basis of matrix vector space

• The standard basis of the matrix vector space $M_{m,n}$ is:

$$(E_{11},E_{12},\dots ,E_{1n},$$

• EX.

  • standard basis of $M_{2,2}$

1 & 0 \\ 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix},\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix},\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$$