Homework

1. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 2. Suppose A is a 4 x 2 matrix with two pivot positions. (a) Does the equation Ax = 0 have a nontrivial solution? Why? (b) Does the equation Ax = b have at least one solution for every possible b? Why? 3. The chemical reaction below can be used in some industrial processes, such as the produc- tion of arsene (AsH3). Use technology to balance the chemical equation MnS + As2Cr10035 + H2SO4 - HMn04 + AsH3 + CrS3012 + H20 Your equation must have coefficients for the chemical reaction as whole numbers. Read the section on Balancing Chemical Equations in Section 1.6 to understand how to set up and solve this type of problem. 4. Suppose A is an m x n matrix with the property that for all b in R" the equation Ax = b has at most one solution. Use the definition of linear independence to explain why the columns of A must be linearly independent