# What is the Definition and Application of Basis Vectors

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!

• How computing an orthonormal basis, e.g. in the case of $C$ a vector in 3 dimensions, look at the Gram-Schmidt process. Apr 24 '20 at 23:17
• You know one of the basic vectors has to be parallel to $C$, so that means it will be of the form $u_1 = \alpha \begin{bmatrix} 4 \\ 5 \end{bmatrix}$ for some scalar $\alpha$. The scalar $\alpha$ should be chosen to make $u_1$ be a unit vector. Then the next basis vector has to be orthogonal to $u_1$, and it also has to be a unit vector. Apr 24 '20 at 23:18
• @littleO, Ah, that makes perfect sense. Thank you for the help! Apr 25 '20 at 0:38

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.