Here is the link to my review topics for exam 1. http://math.uh.edu/~bgb/Courses/Math4377/Math4377-Ex1-Topics.pdf 1.2 Does the solution set to a linear system change under elementary row operations? The solution set does not change under elementary row operations. What are independent/free variables? How can we tell there are free variables by looking at the row-reduced echelon form? I would give an example: 1 0 0 0 0 1 0 0 0 0 0 0 x1=x2=independent, x3=x4=free=linear combination of alpha1 (0 1 0 0) & alpha2 (0 0 0 0) 1.3 How can we rewrite a linear system Ax = b in vector form? Ax=b A= A11 ... A1n . . . Am1 ... Amn x= x1 . . . xn b= b1 . . . bm Can we solve the system if b can be written as a linear combination of the column vectors of A? Yes, I would just put it in augmented form and apply elementary row reductions. I would find that some columns have no pivot variables, thus the variable in that column is free, which is the consequence of being a linear combination of other columns. 1.4 How do the solutions to an inhomogeneous system relate to the solutions of the corresponding homogeneous one? Homogeneous systems are equal to zero a1x1+...anxn=0, thus it is linear independent in which the a1=...=an=0 or it contains only the trivial solution, all xn=0. Inhomogeneous systems are not equal to zero, thus it's solutions will not all be zero. THANKS!!!