1. Terms
Scalar - a single number (Integers, real numbers, rational numbers...)
Vetors - 1-D array of numbers, usually assumed as column(열)
Matrices - 2-D array of numbers, row vectors로 구성됨
Tensors - general form of array of numbers
2. Linear Algebra
- Matrix transpose
$ (A^T)_i,_j = A_j,_i $
$ A = \begin{bmatrix}
A_1,_1 & A_1,_2 \\
A_2,_1& A_3,_2\\
A_3,_1& A_3,_2 \\
\end{bmatrix}
=>
A^T = \begin{bmatrix}
A_1,_1& A_2,_1 &A_3,_1 \\
A_1,_2& A_2,_2 &A_3,_2 \\
\end{bmatrix} $
$ (AB)^T = B^TA^T$
- Matrix multiplication (dot product)
to multiplicate between
# of column from first matrix must be same with the # of row from next matrix
$ C = AB $
$ C_i,_j = \sum_{k}^{} A_i, _k B_k,_j $
if you multiplicate $m \times n $ matrix A and $n \times p$ matrix B the result will be $m \times p$ matrix
- Identity matrix
$A \cdot I = A$
if you do dot product with I matrix the result is same with the previous matrix.
$\begin{bmatrix}
1& 0& 0\\
0& 1& 0\\
0&0 & 1 \\
\end{bmatrix}$
- Matrix Inverse
$A^{-1}A = AA^{-1} = I_n$
- Trace operator
$Tr(A) = \sum_{i}^{}A_{i,i}$
sum of all the diagonal entities of a matrix ( 대각선 원소들의 합)
- Linear Combination
각 벡터 $v^{(i)}$에 해당하는 스칼라 계수를 곱하여 모두 더한것.
$ \sum_{i}C_iv^{(i)}$
- Linear span
linear combination으로 얻을 수 있는 모든 점의 집합
- Linear Independence
a set of vectors is linearly independent if no vector in the set is a linear combination of other vectors