Finding coefficients to express a vector as a linear combination of the basis vectors

• To express $\vec{x}$ as a linear combination of our basis vectors, we start with:

$$\begin{array}{r}B=\{{\vec{v}}_1,...,{\vec{v}}_k\}\\ \\ \vec{x}=c_1{\vec{v}}_1+...+c_k{\vec{v}}_k\end{array}$$

• Normally we would write out the equations and do RREF.
• Here we multiply both sides by ${\vec{v}}_i$, it’s just an arbitrary vector in the orthogonal set.

$$\begin{array}{r}{\vec{v}}_i\cdot \vec{x}={\vec{v}}_i\cdot (c_1{\vec{v}}_1+...+c_k{\vec{v}}_k)\\ =c_1{\vec{v}}_i\cdot {\vec{v}}_1+...+c_k{\vec{v}}_i\cdot {\vec{v}}_k\\ =0+...+c_i{\vec{v}}_i{\vec{v}}_i+...+0\end{array}$$

• Distributing the dot product to all the elements of the LI equation, we see that all the terms except one become zero.
• Since $v_i$ is orthogonal to every vector in the set except $v_i$, all the other dot products zero out.

$$\begin{array}{r}{\vec{v}}_i\cdot \vec{x}=c_i({\vec{v}}_i\cdot {\vec{v}}_i)\\ \\ =c_i||{\vec{v}}_i|{|}^2\\ \\ c_i=\frac{{\vec{v}}_i\cdot \vec{x}}{||{\vec{v}}_i|{|}^2}\\ \end{array}$$

Just for fun:

$$\begin{array}{r}\vec{x}=\frac{{\vec{v}}_1\cdot \vec{x}}{||{\vec{v}}_1|{|}^2}{\vec{v}}_1+\frac{{\vec{v}}_2\cdot \vec{x}}{||{\vec{v}}_2|{|}^2}{\vec{v}}_2+...+\frac{{\vec{v}}_n\cdot \vec{x}}{||{\vec{v}}_n|{|}^2}{\vec{v}}_n\\ \\ \vec{x}=\sum _{i=1}^n{\text{proj}}_{{\vec{v}}_i}(\vec{x})\end{array}$$

Makes sense too, the vector $x$ is the sum of it’s components projected onto the basis vectors.