1. ## unique solution

Does the matrix
0 1 0 -15
0 0 1 7
has a unique solution?

2. ## Re: unique solution

What do you mean by "solving" a matrix?? Is this the augmented matrix for two equations in three unknowns? Well, if so, two equations in three unknowns, no matter what the numbers are, cannot have a unique solution.

3. ## Re: unique solution

"Is this the augmented matrix for two equations in three unknowns?" Yes, it is.
So, does in this case the first (left) unknown is a parameter?

4. ## Re: unique solution

if what you really mean is:

"does

$\begin{bmatrix}0&1&0\\0&0&1 \end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix} = \begin{bmatrix}-15\\7 \end{bmatrix}$

have a unique solution?"

(1,-15,7) is a solution, and so is (2,-15,7), and these are different. there are LOTS of different ways to look at this:

1. we have 3 unknowns, and 2 equations. so IF the equations are consistent (and they are), the system is "under-determined" (more unknowns than equations).

2. the rank of the matrix is 2, which is less than the dimension of the domain.

3. as a rule: general solution = any specific solution + ALL homogeneous solutions. the homogenous case is found by replacing (-15,7) with (0,0) and solving THAT case.

this is equivalent to finding the null space of the matrix. it is clear in this case, that the null space is all vectors of the form (t,0,0) (where t can be any real number, assuming that is the field you are working in).

"free variables" correspond to the COLUMNS without a pivot (leading 1). in this case, that is column 1, so "x1 is free".

let me explain number 2. above a bit more: when a matrix A "acts" on a vector space Rn, by taking x to Ax, it takes an n-vector to an m-vector (supposing that A is mxn). so A(Rn) (the image space of A) is some subspace of Rm. now if m < n (more columns than rows), A takes Rn to a "smaller" space.

you can't take a "bigger" thing to a "smaller" thing without some "collapsing" occuring (two different vectors in Rn have to wind up "with the same image", since Rm doesn't have enough dimensions to reproduce all of Rn faithfully). this is what causes "non-uniqueness" (sorry for all the quotes, but i'm not speaking precisely, here).

what rank(A) tells us, is "how big the image space is". in this case, it has dimension 2: and a basis for im(A) is: {(1,0),(0,1)} (in other words A is of FULL rank, the image of A is all of the target space...A is surjective, or "onto" R2. note that these are precisely the linearly independent COLUMNS of A: the image of A is also known as the COLUMN space).

and THAT means that Ax = b is going to have a solution (in fact, many solutions) for any b, including b = (-15,7).

going back to number 3: since a homogeneous solution is: (t,0,0) for any (real) number t, and (0,-15,7) is a PARTICULAR solution,

the general solution is: (t,0,0) + (0,-15,7) = (t,-15,7), where we are "free" to pick t as we please.

5. ## Re: unique solution

I'm grateful to you for your profound explanation...!!!

If you let me just ask one more question:
according to
in other words A is of FULL rank, the image of A is all of the target space...A is surjective, or "onto" R2.
if A was
010
000
and b was (-15,0)
I should say "the image of A is just R"?

6. ## Re: unique solution

no, because the image of A ISN'T R, it's a 1-dimensional subspace of R2.

these are isomorphic (they "act alike") as VECTOR SPACES, but not as "other things".

for example, in R, we have multiplication, and "fractions". what is:

1/(a,0) ? what does that even MEAN (how do you "divide by a vector")?

7. ## Re: unique solution

Thank you...You're a genius...