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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!

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  • $\begingroup$ How computing an orthonormal basis, e.g. in the case of $C$ a vector in 3 dimensions, look at the Gram-Schmidt process. $\endgroup$ Apr 24 '20 at 23:17
  • $\begingroup$ 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. $\endgroup$
    – littleO
    Apr 24 '20 at 23:18
  • $\begingroup$ @littleO, Ah, that makes perfect sense. Thank you for the help! $\endgroup$
    – Athenian
    Apr 25 '20 at 0:38
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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.

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