Basic linear algebra

This brief tutorial on some key terms in linear algebra is not meant to replace or be very helpful to those of you trying togain a deep insight into linear algebra. Rather, this brief introduction to some of the terms and ideas of linear algebra ismeant to provide a little background to those trying to get a better understanding or learn about eigenvectors andeigenfunctions, which play a big role in deriving a few important ideas on Signals and Systems. The goal of theseconcepts will be to provide a background for signal decomposition and to lead up to the derivation of the Fourier Series .

Linear independence

A set of vectors $\forall x, x_i\in \mathbb{C}^n\colon \{x_1, x_2, \dots , x_k\}()$ are linearly independent if none of them can be written as a linear combination of the others.

Linearly Independent

For a given set of vectors, $\{x_1, x_2, \dots , x_n\}()$ , they are linearly independent if $$c_1{x}_1+c_2{x}_2+\dots +c_n{x}_n=0$$ only when $c_1=c_2=\dots =c_n=0$

As we have seen in the two above examples, often times the independence of vectors can be easily seen through a graph.However this may not be as easy when we are given three or more vectors. Can you easily tell whether or not thesevectors are independent from [link] . Probably not, which is why the method used in the above solution becomesimportant.

Span

Span

The span of a set of vectors $\{x_1, x_2, \dots , x_k\}()$ is the set of vectors that can be written as a linear combination of $\{x_1, x_2, \dots , x_k\}()$ $$\mathrm{span}(\{x_1, \dots , x_k\}())=\{\forall \alpha , {\alpha }_i\in \mathbb{C}^n\colon {\alpha }_1{x}_1+{\alpha }_2{x}_2+\dots +{\alpha }_k{x}_k\}()$$

Basis

Basis

A basis for $\mathbb{C}^n$ is a set of vectors that: (1) spans $\mathbb{C}^n$ and (2) is linearly independent.

If $\{b_1, \dots , b_2\}()$ is a basis for $\mathbb{C}^n$ , then we can express any $x\in \mathbb{C}^n$ as a linear combination of the $b_i$ 's: $$\forall \alpha , {\alpha }_i\in \mathbb{C}\colon x={\alpha }_1{b}_1+{\alpha }_2{b}_2+\dots +{\alpha }_n{b}_n$$

In the two basis examples above, $x$ is the same vector in both cases, but we can express it in many different ways (we give only two out of many, manypossibilities). You can take this even further by extending this idea of a basis to function spaces .