Problem Set 2

See the Guidelines. I will post on ASULearn answers to select questions I receive via messaging there or in office hours.

Problem 1:  1.3 #30

Problem 2: 
Look at Vector([2,-1,1]) and let L=Span of (2,-1,1) = {t (2,-1,1) where t is real}. Notice L is a line through the origin in R3 and we can graph the vector in Maple (without the arrowhead) as:
with(LinearAlgebra): with(plots):
a:=spacecurve([2*t,-1*t,1*t],t=0..1,color=red):
display(a);

Part A: Find a vector w1 so that {Vector([2,-1,1]), w1} spans a plane P1.
Part B: Set up the augmented matrix resulting from the spanning equation (with the vector (b1, b2, b3) as the equals column) and use Gaussian in Maple to find an equation involving b1, b2 and b3 that represents the plane that Vector([2,-1,1]) and w1 lie in.
Part C: Find a vector w2 not on the same line through the origin as w1 so that {Vector([2,-1,1]), w2} also spans the same plane P1.
Part D: In Maple, use spacecurve commands and display as above to show that all three vectors lie in the same plane but no 2 are on the same line [use different colors like black, blue, green..., and one display command like display(a,b,c); and turn the plane "head on" to show this]
Part E: Describe all the vectors w for which {Vector([2,-1,1]), w} spans the same plane P1 as a linear combination
a Vector([2,-1,1]) + b w1 where you specify some condition on a or b or both
[Hint - linear combinations are in the same geometric space so think about what linear combinations
a Vector([2,-1,1]) + b w1
you can use that will still span the plane - and not just a line - ie what a's and b's you can use].
Part F: Find a vector u so that {Vector([2,-1,1]), u} spans a different plane P2 through the origin. Describe how you choose u.
Part G: Add u to your graph from Part D to show it lies outside the plane.

Problem 3: 
Concrete mix, which is used in jobs as varied as making sidewalks and building bridges, is composed of five main materials: cement, water, sand, gravel, and fly ash. By varying the percentages of these materials, mixes of concrete can be produced with differing characteristics. For example, the water-to-cement ratio affects the strength of the final mix, the sand-to-gravel ratio affects the "workability" of the mix, and the fly-ash-to-cement ratio affects the durability. Since different jobs require concrete with different characteristics, it is important to be able to produce custom mixes.

Assume you are the manager of a building supply company and plan to keep on hand three basic mixes of concrete from which you will formulate custom mixes for your customers. The basic mixes have the following characteristics:
  Super-Strong Type S   All-Purpose Type A   Long-Life Type L  
Cement 30 1812
Water 10 1010
Sand 5 2515
Gravel 5 515
Fly ash 10 28
Each measuring scoop of any mix weighs 60g, and the numbers in the table above give the breakdown by grams of the components of the mix. Custom mixes are made by combining the three basic mixes. For example, a custom mix might have 10 scoops of Type S, 14 of Type A, and 7.5 of Type L. We can represent any mixture by a vector [c,w,s,g,f] in R5 representing the amounts of cement, water, sand, gravel, and fly ash in the final mix.

Part A: Give a practical interpretation to the linear combination 3S+5A+2L by comparing the resulting strength (low water to cement ratio), workability (high sand to gravel ratio), and durability (high fly ash to cement ratio) of the mix to the corresponding ratios for S,A and L

Part B: What does Span{S,A,L} = {a S + b A + c L where a, b, and c are real numbers} represent in this context?

Part C: A customer requests 6 kg of (6000 g) of a custom mix with the following proportions of cement, water, sand, gravel, and fly ash: 18:10:19:8:5. Is it possible to make using only S, A and L? If so, find the number of scoops of the basic mixes (S, A, and L) needed to create this mix.

Part D: If there is a solution in Part C, is the solution unique? Explain.

Part E: Let V=Vector([18,10,19,8,5]). Is S, A, L, V linearly independent? Show work using the definition of l.i. and relate your explanation to the definition too.

Part F: Explain why any combination of S, A, L and V can also be achieved by a combination of just S,A, and L, ie that V is redundant.

Part G: Let U=Vector([12,12,12,12,12]). Show that {S,A,L,U} is a linearly independent set of vectors. What practical advantage does that have?

Part H: Define a fifth basic mix W to add to {S,A,L,U} such that any custom mixture can be expressed as a linear combination of the set of mixes {S,A,L,U,W}. Show (and explain) why this works.

Part I: Why will there still be mixes that cannot be physically produced from this set of five basic mixes? Give an example where this happens.

A Review of Various Maple Commands:
> with(LinearAlgebra): with(plots):
> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
> ReducedRowEchelonForm(A);
> GaussianElimination(A); (only for augmented matrices with unknown variables like k or a, b, c in the augmented matrix)
> Vector([1,2,3]);
> A.B;
> A+B;
> B-A;
> 3*A;
> A^3;
> evalf(M) (decimal approximation of M)
> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2); plot vectors as line segments in R3 (ie the columns of matrices) to show whether the the columns are in the same plane, etc.
> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);
> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4); plot equations of planes in R^3 (rows of augmented matrices) to look at the geometry of the intersection of the rows (ie 3 planes intersect in a point, a line, a plane, or no common points)
> display(a,b,c); if plots are set to variables (careful to end those with a colon instead of semi-colon) like a:=implicitplot(...): then this will display them on the same plot