Problem Set 5 Comments and Hints

General comments - be careful in your explanations to specify what systems you are solving for (Ax=0 for linear independence, Ax=[u1,...] for span). The ASULearn group problems and 4.4-4.6 practice solutions are helpful to review. Recall that vectors go in as columns in the corresponding augmented matrices.

4.4 number 16

We want to know whether any vector (u1,u2,u3) in R^3 can be written can be written as a linear combination of the vectors in S. Linear Independence does not tell us anything about this (in fact we know the system is linear dependent, as 4 vectors is too many to efficiently represent R^3), since these inefficient vectors could still represent all of R^3. Instead, we need to either argue in general why every single R^3 vector can be written as a combo or produce a vector in R^3 that cannot be written as such a combination. Use Gaussian on the augmented matrix with u1, u2, and u3 as the 5th column, and reduce to show that there are some choices that give an inconsistent Gaussian reduction. If Gaussian can be inconsistent, then you can find the geometry of the column vectors by examining the algebra of the vector (via setting the inconsistent part =0) that would make the system consistent.

Concrete Application Part 2 (ALL IN MAPLE)

Note that "In Maple" means that you must nicely type all the parts in Maple - text comments too.

Part a Form the augmented matrices Matrix([S,A,L,U]); and Matrix([S,A,L,V]); and then reduce to reduced row echelon form in order to see whether you get a solution. If you get a solution that means that the last vector in the augmented matrix can be written as a linear combination of the 1st three, and so the 4 are not linearly independent. If you don't get a solution, it does not tell you whether the 4 vectors are linearly dependent or independent since one of the first three vectors could still be written in terms of the others and the 4th vector (ie reordering the augmented matrix could give a solution).

Part b Use your answer in part a and what linear dependence means to help you answer the general question and the specific example. Write out the solution of V in terms of the others and substitute.

Part c Follow class notes in order to test for Linear Independence of S, A, L, U by setting up the homogeneous equation and solving to see if there are infinitely many solutions or just one.

Part d Try Z:=Vector([0,0,0,0,60]): We can represent any mixture by a vector [c,w,s,g,f] in R^5 representing the amounts of cement, water, sand, gravel, and fly ash in the final mix, so test out the spanning augmented system on this and use Gaussian to see that it will never be inconsistent.

Part e Think about what would happen in real-life if, when you solve for Matrix([S,A,L,U,Z]) x = b, you obtain negative values for x. Give an example of this happening where b is non-negative (ie an actual real-life mix), but x has at least one negative entry.

4.6 Problems Solve these like in chapter 1, and write out parametrizations, if there are any free variables. Then factor out any free variables...