Note that this asks about general vectors, so do not define the vectors to be ones with specific numbers - leave them general. Do use the general definition of center of gravity as written in #29 and rewrite it as a linear combination. The definition already gives you a version very close to that, but you'll need to do a tiny bit of algebra of vectors and solve for the c_i's.

Look at Vector([1,-2,3]) and let L=Span of (1,-2,3) = {t (1,-2,3) where t is real}. Notice L is a line through the origin in R

with(LinearAlgebra): with(plots):

a:=spacecurve([t,-2*t,3*t],t=0..1,color=red,linestyle=solid):

display(a);

AugmentedPr2:=Matrix([Vector([1,-2,3]),Vector([your vector w1]),Vector([b1,b2,b3])]);

GaussianElimination(AugmentedPr2);

a:=spacecurve([t,-2,3*t],t=0..1,color=red, linestyle=solid): to show that all three vectors lie in the same plane but no 2 are on the same line. (Hint: you can 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-just be sure you have executed with(plots): Another option for plotting is: spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]}, color=red, linestyle=solid); )

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 | 20 | 18 | 12 |

Water | 10 | 10 | 10 |

Sand | 20 | 25 | 15 |

Gravel | 10 | 5 | 15 |

Fly ash | 0 | 2 | 8 |

**Part A**: Compute 3S+5A+2L.

**Part B**: Compare the strength of 3S+5A+2L to that of Types S, A and L - rank
them in order of strength (low water to cement ratio). (Note that this is
one example of a practical interpretation of vector coordinates).

**Part C**: What does Span{S,A,L} = {a S + b A + c L where a, b, and c are
real numbers} represent in this context - relate your answer to the word mixes?

**Part D**: A customer requests a custom mix with the
following proportions of cement, water, sand, gravel, and fly ash:
16:10:21:9:4 .
Is it possible to make using only S, A and L? If so,
find the weights (in scoops/parts of scoops)
of the basic mixes (S, A, and L) needed to create
this mix.
If you choose to do this by-hand, don't forget to show the by-hand work, like row reduction. If you choose to use Maple, you can use commands like:

S := Vector([20,10,20,10,0]):

A := Vector([18,10,25,5,2]):

L := Vector([12,10,15,15,8]):

V := Vector([16,10,21,9,4]);

N := Matrix([S,A,L,V]); ReducedRowEchelonForm(N);

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

**Part F**: Let V=Vector([16,10,21,9,4]). Explain why any linear 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 (think about substitution and using your answer from
part D).

**Part G**: What is the definition of linear independence using vectors? Using a matrix with the vectors as its columns?

**Part H**: Is S, A, L, V linearly independent? Why or why not?

**Part I**: Let U=Vector([12,12,12,12,12]).
Show that {S,A,L,U} is a linearly independent set of vectors by using the
definition of l.i.
If you choose to do this by hand, then show your reduction work and steps. If you choose to use Maple, and have already defined the vectors S, A and L, as in the sample Maple commands in part D, then you can use commands like:

U := Vector([12,12,12,12,12]): zero:= Vector([0,0,0,0,0]):

SALUzero:=Matrix([S,A,L,U,zero]); ReducedRowEchelonForm(SALUzero);

**Part J**: What practical advantage does {S,A,L,U} being linearly
independent have?

**Part K**: Compute the mix S+A+L+U.

**Part L**: Next modify the first entry of the
vector S+A+L+U to
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}. Set up the augmented system {S,A,L,U,W, generic vector} and
use GaussianElimination in Maple to show that your vectors span R^{5}

Once you define W in Maple using a Vector command, you can execute:

SALUWspan:=Matrix([S,A,L,U,W,Vector([b1,b2,b3,b4,b5])]); GaussianElimination(SALUWspan);

**Part M**: In real-life, there will still be mixes that cannot be physically
produced from this set of five basic mixes.
Give an example of a custom mix where at least one weight (of S,A,L,U,W)
is negative and
in your annotations address why we can't use negative weights in mixing cement in real-life.

A Review of Various Maple Commands:

> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);

> ReducedRowEchelonForm(A);

> GaussianElimination(A);

> Vector([1,2,3]);

> A.B;

> A+B;

> B-A;

> 3*A;

> A^3;

> evalf(M)

> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, linestyle=solid);

> 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);

> display(a,b,c);