# True or False

• Feb 19th 2013, 05:11 PM
Civy71
True or False
A linear system of two equations in three unknowns can not have exactly one solution. True or False?
• Feb 19th 2013, 08:24 PM
ibdutt
Re: True or False
Think like this, we can find the values of variables if the number of equations and number of variables is same.
• Feb 19th 2013, 10:27 PM
Deveno
Re: True or False
rather than give a "formal" answer i'll give an "informal one". one can think of the dimension of a vector space as quantifying its "size" in some sense. a system of two equations in 3 unknowns is equivalent to a 2x3 matrix, which takes your inputs (the 3 unknowns) and spits out two numbers (the two equations evaluated at the inputs).

so we have a mapping from a three-dimensional space to a two-dimensional one. this can't happen without at least one dimension "collapsing" down to 0. the dimension(s) that collapses is some line in the three-dimension space that gets mapped to a single point (the origin of the two-dimensional space). this is a many-to-one operation, which is NOT uniquely reversible.

all of this can be formalized by using the concepts of rank, linear independence and spanning set, but the "idea" is what i just said above.
• Feb 20th 2013, 05:43 AM
Hartlw
Re: True or False
Quote:

Originally Posted by Civy71
A linear system of two equations in three unknowns can not have exactly one solution. True or False?

True

xa +yb +zc = b
xa +yb = b-zc
where z can be assigned arbitrarily

a.b,c,d are the columns of the two linear equations. x,y,z are the unknowns
• Feb 20th 2013, 06:16 AM
tejasnatu
Re: True or False
The rank of augmented matrix A:B and A have to be same(=r say) for solution to exist, if r=n unique solution, r<n, infinitely many solutions . In case of 2 equations and 3 variables the rank of augmented matrix and matrix A can be at the most 2 wich is less than 3, so infinitely many solutions exist wich can be found out by putting one unknown equal to a parameter and finding other unknowns in terms of that parameter.
• Feb 20th 2013, 06:44 AM
Hartlw
Re: True or False
ax + by + cz = d is only valid if d is a LC of a,b and c.

rank is less transparent and more wordy (in my opinion).