Yet another proof of Theorem PSPHS

[Valerón]

Since I think I've got a more intuitive proof for this theorem, I'd
like to post it here. But I'm not good at checking for errors, so I
don't know if there's bug in my proof. I hope you can help me to debug
or tell me I did right.

Proof : We say A is a system of m equations in n variable. Let B be a
row-equivalent m * (n + 1) matrix in reduced row-echelon form. Let B'
be the one for LS(A,0). Applying theorem VFSLS, we know that we can
construct any solution of a system by linear combination. For B, here
we use u1,u2,u3,u4....u(n-r) to denote the vectors correspond to the
variable part of the constuction, c as the constant vector. For B', we
use u'1,u'2,u'3,u'4....u'(n-r) as variable part and c' as constant
vector.
Since row-reduce only envoke row operations that make every entry in B
and B' linear relate to other entrys in their columns, so the only
difference between B and B' is the last column, [B](i,n+1) and [B']
(i,n+1), for 1<= i <= m. And [B'](i,n+1) = 0 for all 1<= i <= m.
This make the u1= u'1, u2 = u'2, u3 = u'3....u(n-r) = u' (n-r). And c'
= 0.

Note: The scalers used in the proof are all arbitary chosen.
The first half:
Since w and y are solutions to LS(A,b)
w = c + a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4+....a(n-r) * u(n-r)
y = c + b1 * u1 + b2 * u2 + b3 * u3 + b4 * u4+....b(n-r) * u(n-r)

y - w = (c + b1 * u1 + b2 * u2 + b3 * u3 + b4 * u4+....b(n-r) * u
(n-r))
- (c + a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4+....a(n-r)
* u(n-r))
= (b1 - a1) * u1 + (b2 - a2) * u2 + (b3 - a3) * u3 + (b4 -
a4) * u4..... + (b(n-r) - a(n-r)) * u(n-r)

this show that y - w is a linear combination of u1,u2,u3,u4....u(n-
r), then it's a solution to LS(A,0), N(A) in other words.

Sencond half:
According to the hypothesis and VFSLS
w = c + a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4+....a(n-r) * u(n-r)
z = e1 * u1 + e2 * u2 + e3 * u3 + e4 * u4+....e(n-r) * u(n-
r)
y = w + z
= c + a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4+....a(n-r) * u(n-r)
+ e1 * u1 + e2 * u2 + e3 * u3 + e4 * u4+....e(n-r) * u(n-
r)
= c + (a1 + e1) * u1 + (a2 + e2) * u2 + (a3 + e3) * u3 + (a4 +
e4) * u4 +...... + (a(n-r) + e(n-r)) * u(n-r)
this show that y is a solution to LS(A,b).

Done.

By the way, how to formulate a proof better without TeX, or I have
learn TeX to write proof?
HTML is suck for math proof so we'd better setup a standard for
disccusion in the group.

[Rob Beezer]

Hi Valeron,

I've started a new thread about notation, so it will be easier for
others to find. it's a good idea to address that - thanks for asking.

Can you clarify what you mean by:

For B, here we use u1,u2,u3,u4....u(n-r) to denote the vectors
correspond to the

variable part of the constuction,....

Are the u_i vectors that are columns of B, or maybe rows? Its not
clear what you mean by the "construction" and then what is the
variable part.

Rob

[Valerón]

I'm a bit lazy to write it down...
all there notions are borrow from the proof of VFSLS in the book.
Here is a full definition
Vector Form of Solutions to Linear Systems
Suppose that [A | b] is the augmented matrix for a consistent linear
system LS(A, b) of m equations
in n variables. Let B be a row-equivalent m * (n + 1) matrix in
reduced row-echelon form. Suppose
that B has r nonzero rows, columns without leading 1's with indices F
= {f_1, f_2, f_3, ...... f_n-r, n+1},
and columns with leading 1's (pivot columns) having indices D = {d_1,
d_2, d_3, .......d_r} Define vectors
c, u_j , 1 <= j <= n - r of size n by
......(these all can be copy from the proof of VFSLS, cause I don't
know how to write something is an object in a set and some set is a
subset to another set)

For LS(A,0), the augment matrix is [A | 0] and each notation of the
corresponding object for it should be with a superscript " ' ", for
example, B' means the row-reduce echelon form of [A | 0], c' means the
constant vector.

[Rob Beezer]

Hi Valeron,

Sorry - I didn't pick up on the notation coming out of VFSLS.

Yes, I think this proof is correct (and if not, it would only be minor
changes). In other words, this seems like a nice alternative.

In some ways, it is like the existing proof, since it manipulates
linear combinations, but maybe it is more intuitive. I think
the"arbitary-ness" of the sacalars does have some appeal over the
scalars being entries of w, y and z in the existing proof. I'll think
about making this the proof in the book.

Any comments from others?

Rob

[Zoli]

I vote for the existing proof.

I think the existing proof is the most simple and intuitive one that
is possible. The mathematical background of this theorem consists of
just the notions: <equation>, <addition>, <zero>. The existing proof
reflects this background well by using only these notions.

However the proposed alternative proof uses much more complicated
notions like <gauss elimination> and <number of pivot columns>.

[Rob Beezer]

Maybe Valeron's idea could be used as a "T" exercise that suggests a
certain approach.

Give another proof of PSPHS using Theorem VFSLS by...., etc, etc