# Show that each vector in an n-dimensional vector space can be represented as the summation of its components along the orthonormal basis.

Show that in an n-dimensional vector space V over the universal set with orthogonal basis {$$a_1, a_2,..., a_n$$}, each vector B can be expressed as:

B = $$\frac{a_1}{||a_1||^2}$$ + $$\frac{a_2}{||a_2||^2}$$ +.... + $$\frac{a_n}{||a_n||^2}$$

I tried something along the lines of using B = $$\frac{(B.V)V}{||V||^2}$$ but I couldn't get to the final answer.

• That a vector is the sum of its components along the basis vector is true of any basis—indeed it’s an essential part of the definition of a basis. Are you really asking about the sum of orthogonal projections onto the basis vectors? That’s a different matter.
– amd
May 27, 2020 at 20:26

Let $$B$$ be a vector of $$V$$. Since $$\{a_1,\ldots, a_n\}$$ is a basis for $$V$$, you can find coefficients $$\alpha_\, \ldots, \alpha_n$$ such that $$B=\alpha_1 a_1+\ldots +\alpha_n a_n$$ Note that \begin{align*} \langle B, a_i\rangle&=\langle \alpha_1 a_1+\ldots +\alpha_n a_n, a_i\rangle \nonumber\\ &=\alpha_1 \langle a_1, a_i\rangle +\ldots +\alpha_n \langle a_n,a_i\rangle\nonumber \end{align*} Since the basis is orthogonal, the above expression simplifies as \begin{align*} \langle B, a_i\rangle&=\alpha_i\langle a_i, a_i\rangle\nonumber\\ &=\alpha_i ||a_i||^2\nonumber \end{align*} Hence $$\alpha_i=\frac{\langle B,a_i\rangle}{||a_i||^2}$$