SOLUTION: Demonstrate that vectors (1,0,0),(0,1,0),(0,0,1),(0,1,1) are not an efficient spanning set by showing that an arbitrary vector in R3 can be expressed in more than one way as a line

Question 1095299: Demonstrate that vectors (1,0,0),(0,1,0),(0,0,1),(0,1,1) are not an efficient spanning set by showing that an arbitrary vector in R3 can be expressed in more than one way as a linear combination of these vectors.
I have (a,b,c)=a(1,0,0)+b(0,1,0)+c(0,0,1)+0(0,1,1)
I need one more example with different scalers.
Answer by ikleyn(52779) (Show Source): You can put this solution on YOUR website!
.

Let me make writing more understandable for this problem.


Let  i = (1,0,0) be the first given vector,

     j = (0,1,0) be the second given vector,

and  k = (0,0,1) be the third given vector.


We also have m = (0,1,1) as the fourth given vector.


Let  c = a*i + b*j + c*k  be an arbitrary vector of your 3D space.


Then it has TWO different presentations.


One is this  c = a*i + b*j + c*k.                  (1)


The second is this:  c = a*i + b*m + (c-b)*k.      (2)


      (Check it by considering its by-component parts.)


These two presentations of the same vector give you the example you are looking for.

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