Math 272: Linear Combinations
Names:

Upload completed labs to Moodle by 4pm Monday: https://lms.ats.amherst.edu/mod/assign/view.php?id=354230

Today's lab has been adapted from material in Visual Linear Algebra by Herman and Pepe.

Visualizing linear combinations

The purpose of this section is to give you some experience visualizing linear combinations of vectors. We begin with a picture of the vectors u=(({{2}, {3}})) (in blue) and v=(({{4}, {-1}})) (in red) along with the vector w=3u+2v=(({{14}, {7}})) (in purple):

({{}})

The above grid is based on the vectors u and v. Thus w is the diagonal of the parallelogram whose sides are 3 grid units in the direction of u and 2 grid units in the direction of v; hence w=3u+2v. (The grid also suggests that the weights 3 and 2 can be thought of as the coordinates of w in the coordinate grid determined by u and v.)

Exercise 1.  In the figure below, a parallelogram is shown in yellow.  Express each of the vertices of the parallelogram as a linear combination of the vectors u and v.

({{}})

Exercise 2. What range of values for a and b in the linear combination au+bv will produce the set of points shown in yellow in the figure below?

({{}})

Application: Concrete Mix

Concrete mix, which is used in jobs as varied as making sidewalks and building bridges, is comprised 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, All-Purpose, Long-Life}, { , Type S, Type A, Type L}, {cement, 20, 18, 12}, {water, 10, 10, 10}, {sand, 20, 25, 15}, {gravel, 10, 5, 15}, {fly ash, 0, 2, 8}})}})

Each measuring scoop of any mix weighs 60 grams, and the numbers in the table 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 (R^5) representing the amounts of cement, water, sand, gravel, and fly ash in the final mix. The basic mixes can therefore be represented by the following vectors:

S = {20,10,20,10,0}
A = {18,10,25,5,2}
L = {12,10,15,15,8}

You may use the command RowReduce[A].  The Transpose[A] command is also handy for defining a matrix with given columns: M=Transpose[{S,A,L}] defines a matrix M whose columns are S, A,and L.

Exercise 3.
(a) Give a practical interpretation in this context to the linear combination 3S+5A+2L.  


(b) What does Span{S,A,L} represent in this context?

Exercise 4. Show that {S,A,L} is a linearly independent set of vectors (we want to show that the only solution to xS+yA+zL=0 is the trivial solution x=y=z=0). Note that this has a practical advantage here because it implies the set does not contain any redundant vectors.

Exercise 5.  A customer requests 6 kilograms (6000 grams) of a custom mix with the following proportions of cement, water, sand, gravel, and fly ash: 16:10:21:9:4. Find the amounts of each of the basic mixes (S, A, and L) needed to create this mix.

Suppose we now want two additional custom mixes, U and V:

({{({{ , Type U, Type V}, {cement, 12, 15}, {water, 12, 10}, {sand, 12, 20}, {gravel, 12, 10}, {fly ash, 12, 5}})}})

U = {12,12,12,12,12}
V = {15,10,20,10,5}

Exercise 6. Show that we can make the custom mix V but not the custom mix U from the three basic mixes S,A,L.

Exercise 7. Explain why any combination of S,A,L, and V can also be achieved by a combination of just S, A and L. For example, show how to make the custom mix 3S+4A+2L+3V using only S, A, and L.

Exercise 8.   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}.  (Guess a simple mixture and test whether it works--keep changing your guess until you find something that works.)

Exercise 9.  Why will there still be mixes that cannot be physically produced from this set of five basic mixes?  Give an example of such a mix.  Hint: think about linear combinations with negative coefficients.