Alg1131W10T4 - Unknown right hand side

Monday, 14 May 2018

12:05 AM

Usually we have a system of linear equations with a specific right hand side, such as that we saw in the last topic That involved finding a parabola that passes through the three points [1, 4] , [2, 9] and [—1, 6] But suppose that we had to solve many such problems, of the general form In this case, we can still use row reduction; but we have to be prepared to move the variables bl 7b2 and b3 around; along with the numbers. Let's see what happens: the augmented system would be We use the operation R2 = - 4R1 and R3 = R3 RI to obtain 1 o o 1 O 1 4 1 1 —2 —2 1 -2 O 1 2 —1 1 —3 1 1 1 1 b2 b2 ¯ 4b1 b3-bl At the next stage we could use R3 = R3 — R2 to get 3 b2-4b1 3 where s 3b I-b2+ö3 O If we continue Gaussian elimination until we reach reduced row-echelon form we end up with 1 o o o 1 o o o 1 - äbl + + åb3 —Jbl 1 bl — äb2 + åb3 This tells us that the original system has solutions ö[l] - 5121/3 + 5131/3

 

Note: • the Maple syntax for bl — —b2 + —b3 iSb [1] -1/2*b [2] • you can use the preview button to preview your answer. In particular we observe that @ there is always a unique solution Jb2 O there is a unique solution unless bl — O there is always an infinite number of solutions 1

 

Determine x and y in terms of bl and b2 if The result is x In this case there is @ a unique solution whatever bl and b2 are C) a unique solution as long as bl and b2 satisfy some condition 5 x + 12 y = bl 12 29 y = b2 C) either zero solutions or an infinite family of solutions; depending on the values of bl and b2 Notes: • the Maple syntax for bl — ub2 isb[l] -1/2*b [2] you can use the preview button to preview your answer

 

Machine generated alternative text: Determine y and z in terms of bl, b2 and b3 if The result is In this case there is a @ a unique solution whatever bl and b2 are C) unique solution as long as bl and b2 satisfy some condition y x—3y-3z x—2y-2z 3x-10y-9z - 513] + ö[21 + b[3] C) either zero solutions or an infinite family of solutions; depending on the values of bl and b2 Notes: • the Maple syntax for bl — ub2 — äb3 iSb[1] you can use the preview button to preview your answer

 

Machine generated alternative text: What happened in the last two questions does not always happen' In the following system we row-reduce the augmented matrix Using the somewhat sophisticated row operation R2 —2R1 + 3R2 , we obtain 9 6m — 8y = b2 —12 bl 9 6 -12 Then we will have solutions precisely when the last column is not a leading column, in other words; when —2b1 +3b2 = O. If this condition is satisfied then we have a solution; in fact infinitely many, since @ the left hand side will have a non-leading column, which corresponds to a parameter O the right hand side is no longer a leading column O this automatically happens Similarly the system has a solution precisely when 6m + 9y 8x + 12y -401+352=0

 

Machine generated alternative text: Find the linear condition on the vector b which ensures that the following system has a solution: The condition should be a linear condition, i.e. something of the form 2b1 — 17b2 + 35b3 = 0 . In fact the condition is Note: • the Maple syntax for 2b1 — 17b2 + 35b3 = 0 is you can use the preview button to preview your answer Geometrically this condition represents O O a line in three-dimensional space a general plane in three-dimensional space a plane through the origin in three-dimensional space.

 

 

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