z = a ⋅ x + b ⋅ y + c ⇒ ∑ i = 1 n ( z i − ( a ⋅ x i + b ⋅ y i + c ) ) 2 = ! min . ⇒ a ∑ i = 1 n x i 2 + b ∑ i = 1 n x i y i + c ∑ i = 1 n x i = ∑ i = 1 n x i z i a ∑ i = 1 n x i y i + b ∑ i = 1 n y i 2 + c ∑ i = 1 n y i = ∑ i = 1 n y i z i a ∑ i = 1 n x i + b ∑ i = 1 n y i 2 + cn = ∑ i = 1 n z i ⇒ | ∑ i = 1 n x i 2 ∑ i = 1 n x i ⋅ y i ∑ i = 1 n x i ∑ i = 1 n x i ⋅ y i ∑ i = 1 n y i 2 ∑ i = 1 n y i ∑ i = 1 n x i ∑ i = 1 n y i n | ⋅ | a b c | = | ∑ i = 1 n z i ⋅ x i ∑ i = 1 n z i ⋅ y i ∑ i = 1 n z i | z=a cdot x + b cdot y + c newline drarrow newline sum from{i=1} to{n} (z_i-(a cdot x_i + b cdot y_i+c))^2=csup{!} `min. newline drarrow newline a sum from{i=1} to{n} x_i^2 + b sum from{i=1} to{n} x_i y_i+ c sum from{i=1} to{n} x_i= sum from{i=1} to{n} x_i z_i newline a sum from{i=1} to{n} x_i y_i + b sum from{i=1} to{n} y_i^2+ c sum from{i=1} to{n} y_i= sum from{i=1} to{n} y_i z_i newline a sum from{i=1} to{n} x_i + b sum from{i=1} to{n} y_i^2+ cn= sum from{i=1} to{n} z_i newline drarrow newline left lline matrix{ sum from{i=1} to{n} x_i^2 # sum from{i=1} to{n} x_i cdot y_i # sum from{i=1} to{n} x_i ## sum from{i=1} to{n} x_i cdot y_i # sum from{i=1} to{n} y_i^2 # sum from{i=1} to{n} y_i ## sum from{i=1} to{n} x_i # sum from{i=1} to{n} y_i # n }right rline cdot left lline stack{a#b#c} right rline = left lline stack{sum from{i=1} to{n} {z_i cdot x_i}#sum from{i=1} to{n} {z_i cdot y_i}#sum from{i=1} to{n} z_i} right rline
z = a ⋅ x + b ⋅ y ⇒ ∑ i = 1 n ( z i − ( a ⋅ x i + b ⋅ y i ) ) 2 = ! min . ⇒ | ∑ i = 1 n x i 2 ∑ i = 1 n x i ⋅ y i ∑ i = 1 n x i ⋅ y i ∑ i = 1 n y i 2 | ⋅ | a b | = | ∑ i = 1 n z i ⋅ x i ∑ i = 1 n z i ⋅ y i | z=a cdot x + b cdot y newline drarrow newline sum from{i=1} to{n} {(z_i-(a cdot x_i + b cdot y_i))^2}` = csup{!} `min. newline drarrow newline left lline matrix{ sum from{i=1} to{n} {x_i^2} # sum from{i=1} to{n} {x_i cdot y_i} ## sum from{i=1} to{n} {x_i cdot y_i} # sum from{i=1} to{n} {y_i^2}} right rline cdot left lline binom{a}{b} right rline = left lline binom{sum from{i=1} to{n} {z_i cdot x_i}}{sum from{i=1} to{n} {z_i cdot y_i}} right rline