Let u and v be non-collinear vectors in R^2 . Let w be the orthogonal projection vector of u on v. Consider two statements : (i) Any vector in R^2 can be written as a linear combination of u and v (ii) w can be written as a linear combination of u and v as w=a u+b v , where both a and b are non-zero real numbers.

Step-by-step solution

Given u and v be non-collinear vectors in R^2 and w be the orthogonal projection vector of u on v . Any vector in R^2 can be written as linear combination of u and v because u v for any constant . Hence, u , v set is linearly independent but w can not be written as a u +b v because w . u 0 and W . v 0 , which is contradiction of orthogonal projection concept.