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 M

Cement Mixing:

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);

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)...