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.
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 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
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
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.
Write out the augmented matrix for the system and solve as in Chapter 1.
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.