Change of Basis

Coordinates in a given Basis

Given a subspace $H$ with a basis $\mathcal{B}=\{{\vec{b}}_1,{\vec{b}}_2\}$ and a vector $\vec{x}\in H$, the coordinates of the vector $\vec{x}$ relative to the basis $\mathcal{B}$ are the coefficients $x_1$ and $x_2$ s.t.

$$x_1{\vec{b}}_1+x_2{\vec{b}}_2=\vec{x}.$$

$\vec{x}$ relative to basis $\mathcal{B}$:

$$[\vec{x}{]}_{\mathcal{B}}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$

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