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=Vector([u_1,...,u_n]) for span).

Many of the problems are similar to the ASULearn group problems and 4.4-4.6 practice solutions so those are helpful to review.

4.4 number 15

We want to know whether any vector (u1,u2,u3) in R3 can be written can be written as a linear combination of the vectors in S and if not, what space they do span. Use Gaussian on the augmented matrix with the original vectors as the first three columns and u1, u2, and u3 as the 4th column, and reduce to see if there are some choices that give an inconsistent Gaussian reduction. If so, then check whether the vectors lie on the same line or plane.

4.4 number 54

Linearly independent means the corresponding homogeneous system has only the trivial solution. Set the l.i. equation up with one vector v and notice that there will be at least one condition needed.

Basis problems

Recall that a basis must be l.i. and span. The vectors go in as columns in the corresponding augmented matrices.

Concrete Application Part 2 (ALL IN MAPLE)

Parts of this are similar to the group 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 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 to help you answer the general question and the specific example. Ie 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, but x has at least one negative entry.

4.6 problems

Write out the augmented matrix for the system and solve as in Chapter 1. Then find the solutions and pull out any free variables to form a basis for the homogenous system, with anything not multiplied by a free variable as the particular solution.