The proof of thm 3.3.2

I want to explain the proof of thm 3.3.2 in some detail, because so many people had trouble with the second part of it on Quiz 5, Summer 2002. [The first part was on Quiz 5, Summer 2000]. This is a fairly typical proof that depends on "the main equation" of Ch 3 (see my "main equation page"), but also on some rules of logic.

Rule 1: The contrapositive of an if-then sentence is equivalent to the sentence itself. Example: If x>2 then x² > 4. The contrapositive is If x² £ 4, then x £ 2. These two red sentences mean the same thing, and you can prove one by proving the other. Just pick whichever looks easier. You can start by "Assume x>2" (etc) or by "Assume x² £ 4" (etc).

Rule 2 : A sentence of the form "If A and B and C then D" can be replaced by "Assume A and B. If C then D". That could then be replaced by "Assume A and B. If not D then not C." (using Rule 1).

So, let's look at the theorem and its proof closely.

Thm 3.3.2 : Let v₁, v₂,... v_n be vectors in V. A vector v Î Span(v₁, v₂,... v_n) can be written uniquely as a linear combination (LC) of v₁, v₂,... v_n if and only if v₁, v₂,... v_n are LI.

Proof : The theorem has 4 separate phrases, which I'll label A, B, C and D.

A) v₁, v₂,... v_n are vectors in V
B) v Î Span(v₁, v₂,... v_n) [meaning, it is a LC of them, that v = a₁v₁ + a₂v₂ +... + a_nv_n ]
C) There is no other way to write v as a LC of them.
D) The v₁, v₂,... v_n are L.I.

Part 1: (the " if " part). Here, we assume A and B.We must prove if D then C. We will assume D and prove C. We do that by showing any second L.C. is really the same as the first L.C. This is written pretty clearly in the text, and people seem to get it, so I won't say more about Part 1. It ends with the sentence "Thus the ...." about halfway down.

Part 2: (the "only if " part). Here, we assume A and B again.We want to prove if C then D. But (using Rule 1), we can prove if not D then not C instead. So, we will assume D is false (that the vectors are L.D.) and prove C is false (that there is a second way to write v as a L.C). All this reasoning is abbreviated by the first phrase of Part 2 ,

On the other hand, if v₁, v₂,... v_n are L.D....

The definition of LD comes next .... 0 = c₁v₁ + c₂v₂ +... + c_nv_n , and not all the c's are zero. You'll notice that this equation ("the main eqn") looks a lot like the one assumed in B). That makes them easy to confuse with each other, but also makes them easy to combine, by adding, for example. In fact, adding them finishes the proof, because it produces a second way that v is a LC of the vectors -

v = (a₁+c₁)v₁ + (a₂+c₂)v₂ +... + (a_n+c_n)v_n

This is not the same equation as in B), because at least one c is not zero. So, we know C is false, and we're done.

Most proofs do not spell out the logic in this much detail. You do not have to use abbreviations like A, B and C in your proofs. Use them if it helps you. Or just guide the reader along with phrases like "On the other hand...." or "Conversely..." (which means the same thing). I suggest that you begin every sentence and formula with such a phrase, even if it is just "So....". But pick the phrase carefully. "So....." has a different meaning than "Since...." or "Assume...." or "We claim that .....".

Here's some practice with if-then, and setting up proofs.

Written by S.Hudson, 6/5/02

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