This module introduces linear algebra, DFT, FFT, matrix and vector.
Matrix review
Recall:
 Vectors in
$\mathbb{R}^{N}$ :
$$\forall {x}_{i}, {x}_{i}\in \mathbb{R}\colon x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N1}\\ \end{pmatrix}$$
 Vectors in
$\mathbb{C}^{N}$ :
$$\forall {x}_{i}, {x}_{i}\in \mathbb{C}\colon x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N1}\\ \end{pmatrix}$$
 Transposition:
 transpose:
$$x^T=\begin{pmatrix}{x}_{0} & {x}_{1} & & {x}_{N1}\\ \end{pmatrix}$$
 conjugate:
$$(x)=\begin{pmatrix}\overline{{x}_{0}} & \overline{{x}_{1}} & & \overline{{x}_{N1}}\\ \end{pmatrix}$$

Inner product :
 real:
$$x^Ty=\sum_{i=0}^{N1} {x}_{i}{y}_{i}$$
 complex:
$$(x)y=\sum_{i=0}^{N1} \overline{{x}_{n}}{y}_{n}$$
 Matrix Multiplication:
$$Ax=\begin{pmatrix}{a}_{00} & {a}_{01} & & {a}_{0,N1}\\ {a}_{10} & {a}_{11} & & {a}_{1,N1}\\ & & & \\ {a}_{N1,0} & {a}_{N1,1} & & {a}_{N1,N1}\\ \end{pmatrix}\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ \\ {x}_{N1}\\ \end{pmatrix}=\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ \\ {y}_{N1}\\ \end{pmatrix}$$
$${y}_{k}=\sum_{n=0}^{N1} {a}_{kn}{x}_{n}$$
 Matrix Transposition:
$$A^T=\begin{pmatrix}{a}_{00} & {a}_{10} & & {a}_{N1,0}\\ {a}_{01} & {a}_{11} & & {a}_{N1,1}\\ & & & \\ {a}_{0,N1} & {a}_{1,N1} & & {a}_{N1,N1}\\ \end{pmatrix}$$ Matrix transposition involved simply swapping the rows
with columns.
$$(A)=\overline{A^T}$$ The above equation is Hermitian transpose.
$$A^T_{k, n}=A_{n, k}$$
$$(A)_{k, n}=\overline{A}_{n, k}$$
Representing dft as matrix operation
Now let's represent the
DFT in vectormatrix notation.
$$x=\begin{pmatrix}x(0)\\ x(1)\\ \\ x(N1)\\ \end{pmatrix}$$
$$X=\begin{pmatrix}X(0)\\ X(1)\\ \\ X(N1)\\ \end{pmatrix}\in \mathbb{C}^{N}$$ Here
$x$ is the
vector of time samples and
$X$ is the vector of DFT
coefficients. How are
$x$ and
$X$ related:
$$X(k)=\sum_{n=0}^{N1} x(n)e^{(i\frac{2\pi}{N}kn)}$$ where
$${a}_{kn}=e^{(i\frac{2\pi}{N})}^{(kn)}={W}_{N}^{(kn)}$$ so
$$X=Wx$$ where
$X$ is the DFT
vector,
$W$ is the
matrix and
$x$ the
time domain vector.
$$W_{k, n}=e^{(i\frac{2\pi}{N})}^{(kn)}$$
$$X=W\left(\begin{array}{c}x(0)\\ x(1)\\ \\ x(N1)\end{array}\right)$$ IDFT:
$$x(n)=\frac{1}{N}\sum_{k=0}^{N1} X(k)e^{i\frac{2\pi}{N}}^{(nk)}$$ where
$$e^{i\frac{2\pi}{N}}^{(nk)}=\overline{{W}_{N}^{(nk)}}$$
$\overline{{W}_{N}^{(nk)}}$ is the matrix Hermitian transpose. So,
$$x=\frac{1}{N}(W)X$$ where
$x$ is the time
vector,
$\frac{1}{N}(W)$ is the inverse DFT matrix, and
$X$ is the DFT vector.