singularity

singularity 🔎

• Matrices
  • Ops
    • Addition
    • Multiply w/Scalar
    • Multiply w/Matrix
    • Multiply w/Vector

Matrices

• represent TRANSFORMATIONS that can be applied TO vectors
• ROW x COLS

\[M= \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}\]

Ops

Addition

• as long as they have the same size

\[\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} + \begin{pmatrix} j & k & l \\ m & n & o \\ p & q & r \end{pmatrix} = \begin{pmatrix} a+j & b+k & c+l \\ d+m & e+n & f+o \\ g+p & h+q & i+r \end{pmatrix}\]

Multiply w/Scalar

\[n \cdot \begin{pmatrix} j & k & l \\ m & n & o \\ p & q & r \end{pmatrix} = \begin{pmatrix} n \cdot j & n \cdot k & n \cdot l \\ n \cdot m & n \cdot n & n \cdot o \\ n \cdot p & n \cdot q & n \cdot r \end{pmatrix}\]

Multiply w/Matrix

• AS LONG AS NUMBER OF COLS IN FIRST (LHS)= NUMBER OF ROWS IN SECOND (RHS)

• Unlike numbers, the order of the multiplication matters, even if you’re multiplying together two square matrices that could be multiplied in either order.

• the product is a matrix with LHS ROWS and RHS COLUMNS

\[\begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix} \times \begin{pmatrix} g & h & i \\ j & k & l \\ m & n & o \end{pmatrix}\]

• each row is a vector in LHS, each col a vector in RHS

• \[LHS (2x3)= \begin{pmatrix} (a,b,c)\\ (d,e,f)\end{pmatrix}\]
• \[\times\]
• \[RHS (3x3)= \begin{pmatrix} \begin{pmatrix}g\\h\\i\\\end{pmatrix} \begin{pmatrix}j\\k\\l\\\end{pmatrix} \begin{pmatrix}m\\n\\o\\\end{pmatrix} \end{pmatrix}\]
• \[=\]
• \[\begin{pmatrix} (a,b,c)\cdot(g,h,i),(a,b,c)\cdot(j,k,l),(a,b,c)\cdot(m,n,o)\\ (d,e,f)\cdot(g,h,i),(d,e,f)\cdot(j,k,l),(d,e,f)\cdot(m,n,o) \end{pmatrix}\]
• where each element is basically dot product of corresponding [vectors]

Multiply w/Vector