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Language: en
so let's talk about linear algebra a
little bit because it is such a it's
both a powerful and a beautiful a
subfield of mathematics so what's your
favorite specific topic in linear
algebra or even math in general to give
a lecture on to convey to tell the story
to teach students okay well on the
teaching side so it's not deep
mathematics at all but I I'm kind of
proud of the idea of the four subspaces
there are four fundamental subspaces
which are of course known before long
before my name for them but can you go
through them can you go through the
future I can yes so the first one to
understand is so the matrix is maybe I
should say the matrix what is the matrix
what's a matrix well so we have a like a
rectangle of numbers so it's got n
columns got a bunch of columns and also
got an M rows let's say and the relation
between so of course the columns and the
rows it's the same numbers so there's
got to be connections there but they're
not simple the they're much the columns
might be longer than the rows and
they've all different the numbers are
mixed up first space to think about is
take the columns so those are vectors
those are points in n dimensions
what's the vector so a physicist would
imagine a vector or might imagine a
vector as a arrow you know in space or
the point it ends at in space for me
it's a column of numbers does it you
often think of this is very interesting
in terms of linear algebra the ends of a
vector you think a little bit more
abstract than the how it's very commonly
used perhaps yeah you think this
arbitrary Speight multi-dimensional
right away I'm in high dimensions and in
the room and yeah that's right in the
lecture I tried a so if you think of two
vectors in ten dimensions I'll do this
in class and I'll readily admit that I
have no good image in my mind of a
vector of arrow int n dimensional space
but whatever you can you can add one
bunch of ten numbers to another bunch of
ten numbers so you can add a vector to a
vector and you can multiply a vector by
three and that's if you know how to do
those
you've got linear algebra you know ten
dimensions yeah you know there's this
beautiful thing about math if you look
string theory and all these theories
which are really fundamentally derived
through math yeah but are very difficult
to visualize it yeah how do you think
about the things like a 10 dimensional
vector that we can't really visualize
yeah do you and and yet math reveals
some beauty Oh underlying me yeah
our world in that weird thing we can't
visualize how do you think about that
difference well probably I'm not a very
geometric person so I'm probably
thinking in three dimensions and the
beauty of linear algebra is that is that
it goes on to ten dimensions with no
problem I mean that if you're just
seeing what happens if you add two
vectors in 3d you then you can add them
in 10 D you're just adding the ten
components so so I I can't say that I
have a picture but yet I try to push the
class to think of a flat surface in ten
dimensions so a plane in ten dimensions
and so that's one of the spaces take all
the columns of the matrix take all their
combinations so uh so much of this
column so much of this one then if you
put all those together you get some kind
of a flat surface that I call a vector
space space of vectors and and my
imagination is just seeing like a piece
of paper in 3d but anyway so that's one
of the spaces the nuts space number one
the column space of the matrix and then
there's the row space which is as I said
different but came came from the same
numbers so we got the column space all
combinations of the columns and then
we've got the row space all combinations
of the rows so those are those words are
easy for me to say and I can't really
draw them on a blackboard but I try with
my thick chalk everybody everybody likes
that a railroad chalk and me too I
wouldn't use anything else now
and and then the other two spaces are
perpendicular to those so like if you
have a plane in 3d just a plane is just
a flat surface in 3d then perpendicular
to that plane would be a line so that
would be the null space so we've got two
we've got a column space a row space and
they're two perpendicular spaces so
those four fit together and the in a
beautiful picture of a matrix yeah yeah
it's sort of a fundamental it's not a
difficult idea comes comes pretty early
in 1806 and it's basic
you