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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