And just like that we have arrived at the heart

of Linear Algebra: the concept of linear

independence. Linear independence is so important

that just about every future video will rely on it

and every subsequent topic in Linear Algebra will be directly

or indirectly built upon the concept of linear dependence.

So before I give you a formal definition,

let me motivate it with an example. I will give you a decomposition example

not unlike many that we've considered so far -

but there will be something very different

about this one. So once again it involves geometric vectors

because we always start with geometric vectors when we arrive at a new concept.

It's our key

to intuition and visualization. So given three vectors a, b, and c,

we need to decompose the vector d as a linear combination of a, b, and c.

And maybe you can almost right away see what makes this example

so different from all the ones we've considered until now.

And its the fact that there are more

that one answer! There's more

than one way of doing it. So in anticipation of that

I will write down two more templates and while I'm doing it,

why don't you think up several ways up decomposing

the vector d as a linear combination of vectors a, b, and c

and we'll see if I come up with them

in the same order as you thought of them.

OK, so let me try and guess the first one you thought of:

I think it best to take advantage of a right angle between a and b

(I'm thinking of both a and b as being unit length, c is

at a 45-degree angled both of them and has length

sqrt(2) to so that perfect arrangement). So

d, taking advantage of this right angle, I think can be seen as

2a plus b. I'll bet you most of you

thought of that linear combination first. So let's write it in!

2a + 1b and none of c.

OK, that's one!

And another one can see from the parallelogram rule involving a and b -

excuse me - and and c. This is just a + c

and none of b. So:

1 of a, 0 of b,

and 1c: a + c. Can you see one more?

So let me show you the one that I see most easily:

It won't fit in the shot, but if we take 2c

and come down to d with -b,

that's another way to get d.

as 2c - b. So:

none of a, -1 of b,

and 2c. And you can probably guess,

or you're beginning to see, that there are probably infinitely

many ways of doing this. So why? Why is it

that before there was only one way to find decomposition

(or maybe none!) and now there are perhaps

infinitely many! What is it about

the vectors a, b, and c that causes this? Can you put it in words

or as a mathematical expression? Well, let me give you the reason as I see it!

the reason is that there is a relationship between

a DNC they're not independent

one to be expressed as a linear combination of the other two

that's the relationship among them among their

and that relationship is see

people they'd both be see equal

K plus he

see equal a-plus p

there's a relationship about pay

be and see so that means

that whenever we're looking at a linear combination that produces a vector d

whenever there's in one way or another

a plus p in there we can't take it out

and replace it with see that actually what happened from the first linear

combination to the second

there were two a we went down to one day which took out to be

and instead we've had it see and that was possible

in other words it produces the same result because

are this relationship between

K be and see and that's actually

precisely what we did in the next step we took out another day

with took out another be and we've made up for it with another see

papacy effectively replaces a busby

and now that we've noticed this weekend

come up with it easily come up with infinitely many linear combinations that

will produce T

let's just do it one more time take out one day

1b & Makeup Forever with another see

so minus K mind if

to be plus 3c is most definitely

will once again be the I just can't resist

making one important site comment right now it's one that I make several times

in the past

videos and one that I will probably make many more time

in the future videos I think we're observing

a wonderful interplay here between algebra and geometry

that so central to the mindset of linear algebra has I C

just two minutes ago we came up with these three linear combination

chair metric Lee will look at the picture we judge the relative

arrangement of the vectors AB and C

hand rather easily we came up with the stream any combination

but third excuse me before

and if if would have proven challenger

but then webservice special relationship between two vectors a BNC

weeks prefered algebraically and once that happened

we came up with for 5 but immediately

with infinitely many linear combinations yield

the better D so the more like this store

his pajama tree is beautiful and powerful in its own right

but limited and also for

is powerful in its own right but with ultra-low

would not have been able to come up with even one linear combination

up AT&T York City because the

is not in this relationship but one all server

yielded helping hand to geometry the result

his incredible so these two subjects are powerful on their own

but it's there combination so much more powerful

than the sum of the parts alright now I'm ready to leave here

with a formal definition up a linear independence

and linear dependence a set of vector

in this case we have three but this definition applies to any number of

factors

a set of vectors is linearly dependent

if one of the factors can be expressed

as a linear combination of the rest

set a record a BNC is linearly

dependent because see can be expressed as a linear combination

Ave A&B out of court a can be expressed as a linear combination of PNC

and be can be expressed as a linear combination of the ANC

so any one of those relationships would have qualified this set is linearly

dependent

which you only need walk a set of vectors is linearly dependent

if one of the factors can be expressed as a linear combination

above the rest now linear

independence his opposite up a linear dependence

a set of vectors is linearly

independent if no one of the factors

can be expressed has a linear combination of the rest

once again a set of vectors is linearly

independent if not the vectors

can be expressed has a linear combination of the rest

are him but next year well upped the ante

and will task ourselves with capturing

all possible linear combinations Ave B&C

that produced the well try to do so as a mathematical

expression and that leader to an alternative definition

of a linear dependence family near independence