at 8:38, why do we only talk about z-3x? why dont we also say that y should be equal to 1 and x should be equal to 1-1? Also.. if my x= 0 and y= 1, then we can also find what is z, z will be z=3*1 so z= 3. Why are we not saying that span of these vectors is (0,1,3) ???
The question is the form of the span. Is it a line, plane, or something else? In the bottom row after the last step, all elements must be zero or there is no solution. Hence, z must be equal to 3x or there is no solution. 3x=z is a line in the x-z plane with y filling out the other dimension of the plane. You can't choose x=0 and z=3 because that is not in the solution set. If x=0 then z=0.
Just to add to the excellent answer by @Michael Mott . The other two rows do NOT mean y = 1 and x = 1-1. Those rows just mean that: x = c1 - c2 and y = c2.
@@DupamineThe equations (c1-c2=x,c2=y and z-3x=0) reflect the matrix results. The column vectors in the matrix span the space reflected by these equations. Any vector in the space can be expressed as a linear combination of the two columns. Another way to express it is (1/3z, y, z).
Awesome video! Thanks for the nice explanation!
at 8:38, why do we only talk about z-3x? why dont we also say that y should be equal to 1 and x should be equal to 1-1?
Also.. if my x= 0 and y= 1, then we can also find what is z, z will be z=3*1 so z= 3. Why are we not saying that span of these vectors is (0,1,3) ???
The question is the form of the span. Is it a line, plane, or something else? In the bottom row after the last step, all elements must be zero or there is no solution. Hence, z must be equal to 3x or there is no solution. 3x=z is a line in the x-z plane with y filling out the other dimension of the plane.
You can't choose x=0 and z=3 because that is not in the solution set. If x=0 then z=0.
Just to add to the excellent answer by @Michael Mott . The other two rows do NOT mean y = 1 and x = 1-1. Those rows just mean that: x = c1 - c2 and y = c2.
@@michaelmott3083 Can we say that our span is (c1-c2=x,c2=y and z-3x=0)? I am just confused why are we ignoring x and y
@@ChandreshPant79 Can we say that our span is (c1-c2=x,c2=y and z-3x=0)? I am just confused why are we ignoring x and y
@@DupamineThe equations (c1-c2=x,c2=y and z-3x=0) reflect the matrix results. The column vectors in the matrix span the space reflected by these equations. Any vector in the space can be expressed as a linear combination of the two columns. Another way to express it is (1/3z, y, z).