Currently, I am having a hard time trying to understand what exactly are basis vectors and how to properly apply them in different mathematical scenarios.
Allow me to give an example:
Find an orthonormal basis in which the vector $C = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ is colinear with one of the basis vectors.
With the above prompt, I could then state that:
$$x_1 ' = \begin{pmatrix} \frac{4}{\sqrt{41}} \\ \frac{5}{\sqrt{41}} \end{pmatrix}$$
$$x_2 ' = \begin{pmatrix} -\frac{5}{\sqrt{41}} \\ \frac{4}{\sqrt{41}} \end{pmatrix}$$
However, why is that? How did the above "solution" even appear? Why is is "correct"?
What happens when I start extending my vector $C$ to a $3 \times 1$ matrix? How do I find – for example – $x_2 '$ and $x_3 '$ basis vectors?
Any assistance in helping me understand the definition and method of use of basis vectors would be greatly appreciated. Thank you for reading through my question!
Best Answer
A basis is a set of vectors from which you can construct any vector in your space by taking a linear combination of the basis elements. We say that the basis spans the space.
To find an orthonormal basis so that $(4,5)$ is collinear with one of the vectors in the basis, say let the basis vectors be $(a,b),\,(c,d).$ Then we have that $(a,b)\cdot(c,d)=ac+bd=0,$ and $a^2+b^2=c^2+d^2=1.$ That means the vectors are orthogonal, and of unit norm. Finally, we must have that $(a,b)=\lambda(4,5)$ for some nonzero real $\lambda.$ It suffices to take $\lambda$ so that the given vector is of unit norm. That is, let $$\lambda=\frac{1}{\sqrt{4^2+5^2}}.$$
Since you now know what $a,\,b$ are, substitute in the previous set of three equations to get a quadratic system in $c,\,d,$ which you should be able to solve.