Linear independence = no free variables (pivot in every column; no all zeros)Express one in terms of the other = linear dependenceFree variable = infinitely many solutionsBad row in echelon form = system inconsistent (no solution)Pivot = leading variable in every rowVector form of the solution = divide into pieces (x1, x2, x3...constants)Ax = 0 is a homogeneous linear systemVectors v1…vnspan all of Rmfrom Rm= pivot in every rowMore columns than rows = free variables and linear dependenceOnto = y (transformation) has a solution for every y; pivot in every row; no bad rows; columnsspan R3One to one = y (transformation) has at most one solution; pivot in every column; no pairs ofvectors with the same output; no free variables; columns are linearly independent;Ax = 0 has only solution x = 0Image of T = set of vectors T(x) for all x; set of outputs of TAddition = only for matrices of the same shape; entry by entry (a