Why is orthogonal basis important?

Question

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that special quality of orthogonal basis (extending to orthonormal) that we choose them over non-orthogonal basis?

Answer 1

If $\{v_1, v_2, v_3\}$ is a basis for $\mathbb{R}^3$, we can write any $v \in \mathbb{R}^3$ as a linear combination of $v_1, v_2,$ and $v_3$ in a unique way; that is $v = x_1v_2 + x_2v_2+x_3v_3$ where $x_1, x_2, x_3 \in \mathbb{R}$. While we know that $x_1, x_2, x_3$ are unique, we don't have a way of finding them without doing some explicit calculations.

If $\{w_1, w_2, w_3\}$ is an orthonormal basis for $\mathbb{R}^3$, we can write any $v \in \mathbb{R}^3$ as $$v = (v\cdot w_1)w_1 + (v\cdot w_2)w_2 + (v\cdot w_3)w_3.$$ In this case, we have an explicit formula for the unique coefficients in the linear combination.

Furthermore, the above formula is very useful when dealing with projections onto subspaces.

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Added Later: Note, if you have an orthogonal basis, you can divide each vector by its length and the basis becomes orthonormal. If you have a basis, and you want to turn it into an orthonormal basis, you need to use the Gram-Schmidt process (which follows from the above formula).

By the way, none of this is restricted to $\mathbb{R}^3$, it works for any $\mathbb{R}^n$, you just need to have $n$ vectors in a basis. More generally still, it applies to any inner product space.

Answer 2

The important thing about orthogonal vectors is that a set of orthogonal vectors of cardinality(number of elements of a set) equal to dimension of space is guaranteed to span the space and be linearly independent. If you have not covered this fact in class, you soon will.

As far as your second question goes, there are no prerequisites for linear algebra, apart from elementary mathematics you learn in high school.

Added later:- The main thing is that orthogonality guarantees linear independence, which is rather convenient.