Question 27273: Hi-I'm having troubles with a problem that is based on Mathematica. This problem deals with vectors and linear dependency. Here is what the problem is asking: Below is the table that describes the composition of the three basic mixtures of concrete, Type S, Type A, and Type L. The corresponding vectors for these mixtures and for two custom mixtures, Type U and Type V, have been entered below.
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
Type U Type V
Cement 12 15
Water 12 10
Sand 12 20
Gravel 12 10
Fly Ash 12 5
(a) Show that {S,A,L} is a linearly independent set of vectors. What practical advantage does that have?
(b) Show that we can make the custom mix V but not the custom mix U from the three basic mixes S,A,L. What does this say about the linear independence of the sets {S,A,L,U} and {S,A,L,V}?
(c) 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.
(d) 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}.
(e) Why will there still be mixes that cannot be physically produced from this set of five basic mixes? (Hint: Consider the signs of the scalar weights.) Give an example of such a mix.
I've got parts (a),(b), and (c) figured out but parts (d) and (e) I'm confused about.
On part (d) I added the four vectors {S,A,L,U} together to get W. After that I put the vectors {S,A,L,U,W,{0,0,0,0,0}} into a matrix and solved for the system. As the solution set shows I get {-c5,-c5,-c5,-c5,c5} for my scalar weights. I set c5=1 and get the linear dependence relation: -S-A-L-U+W=0. I thought this was the right method to define W, but then I realized that you cannot have negative values for the mixes.
Because I cannot get part (d) I'm having a lot of trouble answering part (e). Is there a clearer explanation for the question being asked in part (e)?
If you could help me out with this problem I will greatly appreciate it. Thank you so much for taking the time to help.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! FIRST OF ALL LET ME CONGRATULATE YOU ON YOUR EXCELLENT INSIGHT IN TO THESE PROBLEMS OF LINEAR DEPENDENCE/INDEPENDENCE AND GOOD SHOW IN SOLVING THIS PROBLEM INVOLVING 5 VARIABLES.IN FACT YOU GOT ALL THE ANSWERS ,BUT JUST NOT ABLE TO GIVE A FINISHING TOUCH TO COMPLETEC THE JOB.YOU ARE AT THE END AND ONLY YOU NEED TO REMOVE A SMALL OBSTACLE..SEE BELOW..
Hi-I'm having troubles with a problem that is based on Mathematica. This problem deals with vectors and linear dependency. Here is what the problem is asking: Below is the table that describes the composition of the three basic mixtures of concrete, Type S, Type A, and Type L. The corresponding vectors for these mixtures and for two custom mixtures, Type U and Type V, have been entered below.
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
Type U Type V
Cement 12 15
Water 12 10
Sand 12 20
Gravel 12 10
Fly Ash 12 5
(a) Show that {S,A,L} is a linearly independent set of vectors. What practical advantage does that have?
(b) Show that we can make the custom mix V but not the custom mix U from the three basic mixes S,A,L. What does this say about the linear independence of the sets {S,A,L,U} and {S,A,L,V}?
(c) 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.
(d) 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}.
(e) Why will there still be mixes that cannot be physically produced from this set of five basic mixes? (Hint: Consider the signs of the scalar weights.) Give an example of such a mix.
I've got parts (a),(b), and (c) figured out
VERY GOOD...
but parts (d) and (e) I'm confused about.
WELL..THINK COOLLY..DONT GET CONFUSED.. THERE ARE 5 VARIABLES CEMENT,WATER,SAND,GRAVEL AND FLY ASH..SO YOU NEED 5 VECTORS TO COMPLETELY COVER THE GROUND..YOU ALREADY PROVED THAT S,A,L,U ARE LINEARLY INDEPENDENT..AND V IS NOT INDEPENDENT BUT DEPENDENT ON S,A ANDL . SO THERE WILL BE ALWAYS ANOTHER W SAY AN INDEPENDENT VECTOR.IT CAN BE ANY WHICH IS INDEPENDENT OF S,A,L,V..YOU SAY YOU FOUND IT OUT TOO..GOOD...SO THAT COMPLETES THE D PART...MATHEMATICALLY...YOU CAN HAVE ANY W SO THAT
IF ......T*S+U*A+V*L+X*U+Y*W=0...THEN T=U=V=L=X=Y=0..NOTE THE WORD
MATHEMATICALLY.....THAT IS YOU CAN FIND ANY OTHER CUSTOM MIX SAY Z AS A LINEAR COMBINATION OF S,A,L,U AND W AGAIN MATHEMATICALLY..WHICH MEANS THAT ANY ONE OR MORE COEFFICIENTS OF THESE INDEPENDENT VECTORS COULD BE NEGATIVE!!!!!YA!!THAT IS THE KEY!!THEY CAN BE NEGATIVE IN MATHS!!BUT NOT IN PHYSICALLY!!!!.THIS PROBLEM HAS ARISEN BECAUSE WE HAVE NOT REPEAT NOT IMPOSED THE CONDITION THAT ALL THE COEFFICIENTS HAVE TO BE ZERO OR POSITIVE..THEY CAN NOT BE NEGATIVE..THAT IS THE REASON WHY WE MAY STILL NOT BE ABLE TO MAKE CERTAIN MIXES EVEN THOUGH WE GOT 5 INDEPENDENT VECTORS ,BECAUSE OF THE NECESSITY OF USING NEGATIVE QUANTITIES WHICH IS PHYSICALLY IMPOSSIBLE.
On part (d) I added the four vectors {S,A,L,U} together to get W. After that I put the vectors {S,A,L,U,W,{0,0,0,0,0}} into a matrix and solved for the system. As the solution set shows I get {-c5,-c5,-c5,-c5,c5} for my scalar weights. I set c5=1 and get the linear dependence relation: -S-A-L-U+W=0. I thought this was the right method to define W, but then I realized that you cannot have negative values for the mixes.
THAT IS IT YOU GOT THE ANSWER , BUT JUST NOT ABLE TO FINISH IT OFF....ALSO NOTE THE HINT ON THE TERM 'PHYSICALLY' AND 'NEGATIVE SIGNS'IN THE PROBLEM.
HOPE IT IS CLEAR TO YOU NOW..OTHERWISE PLEASE COME BACK....
Because I cannot get part (d) I'm having a lot of trouble answering part (e). Is there a clearer explanation for the question being asked in part (e)?
If you could help me out with this problem I will greatly appreciate it. Thank you so much for taking the time to help.
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