Assume we have two vectors of order 10, X 1 and X 2 . Jointly these two vectors define a plane, a 2-dimensional subspace of the original 10- dimensional space. Let Z 1 and Z 2 be an arbitrary...

Assume we have two vectors of order 10, X₁
and X₂. Jointly these two vectors define a plane, a 2-dimensional subspace of the original 10- dimensional space. Let Z₁
and Z₂
be an arbitrary coordinate system for this 2-dimensional subspace. Represent the vectors X₁
and X₂
in this plane by the coordinates of the two vectors Z₁
= (5 2) and Z₂
= (0 −4) . Suppose the projection of Y onto this plane plots at (−1 3) in this coordinate system.

(a) Use your figure to approximate the regression coefficients for the regression of Y on X₁
and X₂.

(b) From your figure compute the sum of squares due to the regression of Y on X₁
and X₂
jointly. How many degrees of freedom does this sum of squares have?

(c) Do you have enough information to compute the residual sum of squares? How many degrees of freedom would the residual sum of squares have?

(d) Suppose someone told you that the original vector Y had length 3. Would there be any reason to doubt their statement?