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.

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. Or try a number of different vectors in R^3 to find one that is not in the Span of the set, and then write up only that one...

VLA Concrete Application Part 2 (ALL IN MAPLE)

Parts of this are similar to the groups problems we worked on, and the solutions are on ASULearn. Note that "In Maple" means that you must nicely type all the parts in Maple - text comments too.

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

Part b This is similar to Problem Set 4. 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 c Use your answer in part b to help you answer the general question and the specific example.

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. Explain that since R^5 has dimension 5, we know that it can be represented by a basis consisting of 5 linearly independent vectors. Next show that the set of 5 vectors is linearly independent by setting up the homogeneous system and solving.

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, but x has at least one negative entry.