Okay everyone... Got a question that I can\'t for the life of me understand. Looking for the methodology on how to solve it... Thanks in advance! (Copied from Linear Algebra and It\'s Applications by David C. Lay) Section 1.4, Exercise #34 Let A be a 3 x 4 matrix, let v(1) and v(2) be vectors in R^3, and let w = v(1) + v(2). Suppose v(1) = Au(1) and v(2) = Au(2) for some vectors u(1) and u(2) in R^4. What fact allows you to conclude that the system Ax = w is consisten? (Note: u(1) and u(2) denote vectors, not scalar entires in vectors). ~~~~ Specific questions I have when trying to understand this problem is... At one point we are dealing with vectors with R^3 and then the next part deals with R^4... how do each relate to one another? From what I gather from the definition earlier in the book, A is an m x n matrix and x is in R^n ... so when it refers to Ax = w, is it referring to the 3 x 4 matrix denoting R^3, or is it referring to the matrix with u(1) and (2) denoting R^4. Why? Thanks again! Solution every matrix corresponds to a linear transformation and vice verse The matrix A is 3X 4 matrix thus represent a linear transformation from R^4 to R^3 Thus u(1) and u(2) are vectors in R^4 and V(1) and v(2) are vectors in R^3 now the system is AX = v(1) + v(2) this is a system in 4 variables and there are 3 equations now it is given that Au(1) = v(1) Au(2) = v(2) adding we get Au(1) + Au(2) = v(1) + v(2) => A(u(1) + u(2) ) = v(1) +v(2) [ as matrices over reals form a vector space and distributive law holds} => the system AX = w has a solution.