SELECT HINTS for PS 4

4.1 36 and 44. Recall the definition of linear combination - we set up the system where the columns of the coefficient matrix M are the column vectors: [u1,u2,u3]=M, and Mx=b, where b is the vector you wish to know whether can be written as a linear combination. Then solve this system any way you like for the unknown variables in the column vector x, and relate your answer to what the problem originally asked for. See practice problem solutions 4.1 35 and 43 up on ASULearn for similar problems and a review of various methods for solution, although you will just choose one for your writeup.

Cement Mixing: Note that "In Maple" means that you must nicely type all the parts (a through d) in Maple - text comments too.

You can create column vectors by using
with(LinearAlgebra):
u:=Vector([1,2,3,4,5]):

You can create a matrix of defined column vectors u, v and w and then reduce that matrix by typing it in directly, as usual, or by using
M:=Matrix([u,v,w]):
ReducedRowEchelonForm(M);

Part A Given the proportions of Cement, Water, Sand, Gravel and Fly Ash that you obtain from calculating 3S+5A+2L, discuss the strength (low water to cement ratio), workability (high sand to gravel ratio), and durability (low fly ash to cement ratio) of the mix. For each criteria (like strength) compute the ratio and compare it to the ratio of that given in S, A, and L. (So, for strength, is the mix as strong or stronger as superstrong S or is it weaker like A, or even weaker like L?)
Part B The Span{S,A,L} represents ALL linear combinations of S, A and L, so this represents all of the mixes that are combinations of these three basic mixtures. By changing the coefficients, we can obtain mixes of varying workability, durability, and strength. (You will explore more on this in the next problem set.)
Part C We are asking whether the given custom mix can be written as a linear combination of the column vectors S, A, and L. Recall that if we form the system Mx=b, where M is the matrix whose columns are [S, A, L], and b is the custom mix, then we can solve the the unknown variables in x. Since M is a 5x3 matrix, and b is a 5x1 vector, we know that the mixing matrix M will act on a 3x1 column vector, and by matching up the units we can see that it will have the entries [number of scoops of Type S, number of scoops of Type A, and number of scoops of Type L], (For example, if we look at the first row of M acting on x and giving us the first entry in the 5x1 column vector, we see that the amount (in grams) of cement per scoop of Type S *x1 + the amount (in grams) of cement per scoop of Type A *x2 + the amount (in grams) of cement per scoop of Type L *x3 = the amount of cement (in grams) in the mixture we create, and so x1 is the number of scoops of S, x2 is the number of scoops of A, and x3 is the number of scoops of L.) a good check for M is that each column of M should contain 60 grams, since that is the weight of a measuring scoop.
Here we want 6000 grams of a custom mix with the proportions of cement, water, sand, gravel, and fly ash: 16:10:21:9:4. Notice that there are 60 grams in this mix (by adding 16+10+21+9+4), and so if we want 6000 grams, then we must multiply the proportions by 100. Then we want the amounts of each of the basic mixes (S, A, and L) needed to create this mix which has 1600 grams of cement, 1000 grams of water, 2100 grams of sand, 900 grams of gravel and 400 grams of fly ash. Thus, we want to solve for the 3x1 vector so that M times the vector = the 5x1 column vector with the entries (1600,1000,2100,900,400). Since M is a 5x3 matrix, we cannot use the inverse matrix method (nor Cramer's method) to solve, and so we MUST use the augmented matrix method, which works for any number of equations and unknowns. Then relate your answer to what the problem originally asked, and also answer part D.

4.2 22 and beyond: You may wish to review class notes first. For all of these problems, if it is a vector space/subspace, just say it is, but if not, write out a complete proof, ie, what violating it means, and where each step follows logically from the previous step, like in class. Some more specific comments for some of them:

Subset of R^3 - Solutions to 2x-3y+4z=5. What are the vectors here -- they are {(x,y,z) so that x, y, and z satisfy the equation 2x-3y+4z=5}. You are directed to use axiom 1. That means that you need to produce 2 specific 3x1 column vectors (with real numbers) that can be plugged into the equation 2x-3y+4z and result in a 5. Then continue following the method for disproving axiom 1. Be sure to write your response as a proof.

4.3 number 14 part D. The problem says that we are looking at nxn singular matrices (ie nxn matrices with determinant 0 who have no inverse). The problem specifies a general n, so it does not suffice to define n=2. You may wish to look at examples for n=2, n=3, n=4, n=5, and then go back to a general n. Your final answer will use things like the nxn matrix of all 0s, the nxn identity matrix with 1s along the diagonal and 0s everywhere else, or a similar nxn matrix with general instructions you can give (like the nxn matrix with 0s everywhere except the a11 spot, where a11=1)...