8/10/2019 Proof of Linear Independence

http://slidepdf.com/reader/full/proof-of-linear-independence 1/2

Linear IndependenceHow do we prove it?

Consider the following if-then statement:

“If  A  is an  m × n matrix and  Av1,Av2, . . . ,Av p  are linearlyindependent, then  v1,v2, . . . ,v p  are linearly independent.”

This is a true statement, but how do we go about proving it? To figure this out, let’s analyze the it a littlemore closely.

The hypothesis of the implication is the statement

A  : “A  is an  m × n matrix and  Av1, Av2, . . . , Av p  are linearly independent.”

and the conclusion isD   : “v1,v2, . . . ,v p  are linearly independent.”

As usual, to prove “If   A then   D”, we start by assuming that   A is true and then deduce that   D  is also true.What makes things challenging in this case is the nature of the statement   D: it is also an if-then statement!Indeed, the definition of linear independence tells us that   D   is really the statement

D   : “If  x1v1 + x2v2 + · · · + x pv p  =  0  then  x1 =  x2  =  · · · =  x p  = 0.”

In other words, the conclusion of our original if-then statement is another if-then statement with its ownhypothesis and conclusion! In this case the hypothesis of   D   is

B   : “x1v1 + x2v2 + · · · + x pv p  =  0.”

and the conclusion of   D   is

C   : “x1  =  x2  =  · · · =  x p  = 0.”So the statement we are really trying to prove is the “compound” statement

“If   A then (if   B  then   C )”

or, expressed entirely in symbols,A ⇒ (B ⇒ C ).

How in the world do we prove something like this?1 Very carefully.The easiest thing to do is to replace the expression   A ⇒  (B ⇒ C ) with one that is logically equivalent

but easier to understand. It turns out that the statement   A ⇒ (B ⇒ C ) is logically equivalent to

(A and   B )  ⇒ C .

You can prove this rigorously using a truth table, but it should be intuitively clear that this makes sense.

We can read  A ⇒ (B ⇒ C ) as “Whenever  A  is true, then whenever  B  is true  C   is true.” This certainly seemsto be the same as the statement “Whenever   A and   B  are true, then   C  is true,” which is how we would read(A and   B )  ⇒ C .

So now we see that the statement   A ⇒ (B ⇒ C ) can be proven by assuming   A  and   B  are true and thendeducing that   C   is true. This is because that is exactly how we would prove (A and   B )   ⇒ C , which is alogically equivalent statement.

Summary:   To prove the statement   A ⇒   (B ⇒ C ), start by assuming that   A   and   B   are true, and thendeduce that   C  is true.

Let’s see how we would apply this to our original problem. We want to prove that

1The statement  A ⇒ (B ⇒ C) is  not  the same as the string of implications  A ⇒ B ⇒ C  because there is no “associative law”

in logic that would allow us to shift parentheses (or omit them). In otherwords, we   are not  trying to prove that   A  implies   B

and that  B  implies  C .

8/10/2019 Proof of Linear Independence

http://slidepdf.com/reader/full/proof-of-linear-independence 2/2

“If  A  is an  m × n matrix and  Av1, Av2, . . . , Av p  are linearlyindependent, then  v1,v2, . . . ,v p  are linearly independent.”

is a true statement. We showed above that this is of the form   A ⇒ (B ⇒ C ) where   A,B ,C   are

A   : “A  is an  m × n  matrix and  Av1, Av2, . . . , Av p  are linearly independent.”

B    : “x1v1 + x2v2 + · · · + x pv p  =  0.”

C    : “x1  =  x2  =  · · · =  x p  = 0.”

So, according to what we discussed above, we start by assuming that A is an m×n matrix and Av1,Av2, . . . , Av p

are linearly independent, and  x1v1+x2v2+· · ·+x pv p  =  0. We need to conclude that x1  =  x2 =  · · · =  x p  = 0.Let’s get to it!

Sincex1v1 + x2v2 + · · · + x pv p  =  0

if we multiply both sides by  A  we get, by linearity of matrix/vector multiplication,

A(x1v1 + x2v2 + · · · + x pv p) = A0x1Av1 + x2Av2 + · · · + x pAv p  =  0.

Linear independence of  Av1, Av2, . . . , Av p  tells us that we must have  x1  =  x2  =  · · · =  x p  = 0. This is whatwe needed to show.

Most linear independence proofs follow this same pattern. We are usually asked to prove a statement of the form

“If   A, then  v1,v2, . . . ,v p  are linearly independent.”

The same reasoning as above shows that if   B  is the statement “x1v1 + x2v2 + · · · + x pv p  =  0” and   C   is thestatement  x1   =  x2   =   · · ·  =  x p   = 0, then we are really being asked to prove   A ⇒  (B ⇒ C ). And now weknow how to do this!

Summary:  To prove the statement

“If   A, then  v1,v2, . . . ,v p  are linearly independent.”

start by assuming that   A   and   x1v1  +  x2v2  +  · · ·  +  x pv p   =   0   are true. Then conclude from this thatx1 =  x2  =  · · · =  x p = 0.

Let’s finish by applying this technique to another example. In the proof I’m not going to explicitly pointout that we’re using this method, so be sure you can identify it.

Example.   Let  V   be a finite-dimensional vector space with basis   B  and let  v1,v2, . . . ,v p   ∈ V . Show thatif the   B -coordinate vectors [v1]B, [v2]B, . . . , [v p]B  are linearly independent then so are the original vectors

v1,v2, . . . ,v p.

Proof.  Suppose that [v1]B, [v2]B, . . . , [v p]B  are linearly independent and that

c1v1 + c2v2 + · · · + c pv p  =  0.   (1)

Applying the coordinate map to this expression and using linearity gives

[c1v1 + c2v2 + · · · + c pv p]B   = [0]B

c1[v1]B + c2[v2]B + · · · + c p[v p]B   =   0.

Since [v1]B, [v2]B, . . . , [v p]B  are linearly independent, this can only happen if  c1  =  c2  =   · · · =  c p   = 0. Thatis, (1) can only hold if  c1  =  c2  =  · · · =  c p  = 0, which shows that  v1,v2, . . . ,v p  are linearly independent.