Least Squared Error Solution

Squared error

Squared distance is also known as squared error. Consider a linear equation in $w_i$'s:

$$w_1a_1+w_2a_2+...+w_na_n=b$$

The squared error (squared distance) on a given point, $(\hat w_1,\hat w_2,...,\hat w_n)$, is defined as:

$$SE(\hat w_1,\hat w_2,...,\hat w_n)=(\hat w_1a_1+\hat w_2a_2+...+\hat w_na_n-b)^2$$

Note: In the case of $w=\hat{w}$, the sum of squared errors=0. This implies that we’re able to find an exact solution.

Sum of squared errors

Consider a linear system with $m$ equations and $n$ unknowns and the corresponding squared errors on a point, $(\hat w_1,\hat w_2,...,\hat w_n)$:

Linear System	Sum of Squared Distances
$(1):w_1a_{11}+w_2a_{12}+...+w_na_{1n}=b_1$$(2):w_1a_{21}+w_2a_{22}+...+w_na_{2n}=b_2$$\;\ \;\ \;\ \;\ \;\ \;\ \;\ \;\ \vdots$$(m):w_1a_{m1}+w_2a_{m2}+...+w_na_{mn}=b_m$

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