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Chapter 5

Chapter 5 - F-2011 STAT 510 Chapter 5 Matrix approach to...

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F-2011 STAT 510 1 Chapter 5 Matrix approach to simple linear regression For linear model: x y 1 0 n Y Y Y 2 1 = n n X X X 2 1 1 0 2 1 1 1 1 1 Let n n Y Y Y 2 1 1 Y be the vector of n observations on the response variable and let n n X X X 2 1 2 1 1 1 1 X of the predictor variable X be the design matrix . Then, the regression model can be written as 1 1 2 2 1 n n n ε β X Y 1 2 2 1 } { β X Y n n E Here, 1 n ε is a vector of independent normal RVs with 0 ε } { E and I ε 2 2 } { . Then, the product 2 2 2 2 2 1 2 1 2 1 i n n n Y Y Y Y Y Y Y Y Y Y Y Y

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F-2011 STAT 510 2 2 2 1 2 1 1 1 1 1 1 1 1 i i i n n X X X n X X X X X X X X i i i n n Y X Y Y Y Y X X X 2 1 2 1 1 1 1 Y X Linear dependence and rank matrix Consider the following matrix: 1 6 1 15 10 5 4 2 2 3 2 1 A Are the columns of A linearly independent?
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