CNYT

Note: In this course, i used GIT for version control, GIT HUB for pushing or pulling files and PYTHON for the programming.

First Term

• 1 week : We learned how the complex numbers exit, how they have arithmetic opertation and how they also share the caracteristics with the vectors.

  • Add $(a,b) + (c,d) = (a+c,b+d)$
  • Multiplication $(a,b) * (c,d) = (ac - bd, ad + bc)$
  • Substration $(a,b) - (c,d) = (a - c, b - d)$
  • Division $\frac{(a,b)}{(c,d)} = (\frac{ac + bd}{c^2 + d^2}, \frac{bc - ad}{c^2 + d^2})$
  • Module $|(a,b)| = \sqrt{a^2 + b^2}$
  • Conjugate $(a,b) => (a,-b)$
  • Coordinates
    • Cartecian $(a,b) = a + bi$
    • Polar $(r,θ)$
  • Phase $θ = arctan(\frac{a}{b})$

• 2 week : We learn complex vector and matrix and their propierties. Magnitud, dirección y sentido.

  • vectors

    • add $\rightarrow$ (A+B)[i] = A[i]+B[i]
    • multSca $\rightarrow$ c*A[i]
    • $V+W = W+V$
    • $(V+W)+X = V+(W+X)$
    • $V+O = O+V$
    • $V+(-V) = (-V)+V = O$
    • $1*V = V$
    • $C_1.(C_2.V) = (C_1*C_2).V$
    • $C.(C+W) = C.V + C.W$
    • $(C_1+C_2).V = C_1.V + C_2.V$

  • matrices

    • add $\rightarrow$ (A,B)[j,k] = A[j,k]+B[j,k]
    • neg $\rightarrow$ (-A)[j,k] = -A[j,k]
    • multSca $\rightarrow$ (p.A)[j,k] = p*A[j,k]
    • The same propierties of vectors plus:
    • transpose $\rightarrow$ $(A^2)$[j,k] = A[k,j]
    • conjugate $\rightarrow$ $\bar{A}$[j,k]
    • adjoint $\rightarrow$ $A^t$ = $\bar{(A^T)}$ = $(\bar{A})^T$

  • Basis and Dimension

    • Linear combination $\sum{c.V_n} = V_t$
    • Linearly indepent $\sum{c.V_n} = O$
    • Canonical basis

• 3 week :