선형결합

From BoostCourse 주재걸

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Given vectors $v_1, v_2, \cdot\cdot\cdot, v_p \in \mathbb{R}^n$ and given scalars $c_1, c_2, \cdot\cdot\cdot, c_p$,

$c_1v_1 + \cdot\cdot\cdot + c_pv_p$

is called a linear combination of $v_1, \cdot\cdot\cdot, v_p$ with weights or coefficients $c_1, \cdot\cdot\cdot, c_p$.

 

 

Given a set of vectors $v_1, \cdot\cdot\cdot, v_p \in \mathbb{R}^n$,

Span ${v_1, \cdot\cdot\cdot, v_p}$

is defined as the set of all linear combinations of $v_1, \cdot\cdot\cdot, v_p$.

 

 

When does the solution exist for $Ax = b$?

The solution exists only when $b \in$ Span ${a_1, a_2, \cdot\cdot\cdot, a_n}$.

 

 

1. Matrix Multiplications can be interpreted as Linear Combinations of Vectors
- A linear combination of columns of the left matrix.

$\begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}1 + \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}2 + \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}3$

- A linear combination of rows of the right matrix.

$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix}1 + \begin{bmatrix} 1 & 0 & 1 \end{bmatrix}2 + \begin{bmatrix} 1 & -1 & 1 \end{bmatrix}3$

 

 

2. Sum of (Rank-1) Outer Products

example:

$\begin{bmatrix} 1 & 1 \\ 1 & -1 \\ 1 & 1\end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix}\begin{bmatrix} 1 & 2 & 3\end{bmatrix} + \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\begin{bmatrix} 4 & 5 & 6\end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{bmatrix} + \begin{bmatrix} 4 & 5 & 6 \\ -4 & -5 & -6 \\ 4 & 5 & 6 \end{bmatrix}$

used for covariance matrix in multivariate Gaussian or Gram matrix in style transfer