18.06 Linear Algebra, Spring 2010
Transcript – Lecture 30
OK, this is the lecture on linear transformations. Actually, linear algebra courses used
to begin with this lecture, so you could say I'm beginning this course again by talking
about linear transformations.
In a lot of courses, those come first before matrices. The idea of a linear
transformation makes sense without a matrix, and physicists and other -- some
people like it better that way. They don't like coordinates.
They don't want those numbers. They want to see what's going on with the whole
space. But, for most of us, in the end, if we're going to compute anything, we
introduce coordinates, and then every linear transformation will lead us to a matrix.
And then, to all the things that we've done about null space and row space, and
determinant, and eigenvalues -- all will come from the matrix.
But, behind it -- in other words, behind this is the idea of a linear transformation. Let
me give an example of a linear transformation. So, example.
Example one. A projection. I can describe a projection without telling you any
matrix, anything about any matrix. I can describe a projection, say, this will be a
linear transformation that takes, say, all of R^2, every vector in the plane, into a
vector in the plane. And this is the way people describe, a mapping. It takes every
vector, and so, by what rule? So, what's the rule, is, I take a -- so here's the plane,
this is going to be my line, my line through my line, and I'm going to project every
vector onto that line. So if I take a vector like b -- or let me call the vector v for the
moment -- the projection -- the linear transformation is going to produce this vector
as T(v). So T -- it's like a function.
Exactly like a function. You give me an input, the transformation produces the
output.
So transformation, sometimes the word map, or mapping is used. A map between
inputs and outputs. So this is one particular map, this is one example, a projection
that takes every vector -- here, let me do another vector v, or let me do this vector
w, what is T(w)? You see? There are no coordinates here.
I've drawn those axes, but I'm sorry I drew them, I'm going to remove them, that's
the whole point, is that we don't need axes, we just need -- so guts -- get it out of
there, I'm not a physicist, so I draw those axes. So the input is w, the output of the
projection is, project on that line, T(w). OK.
Now, I could think of a lot of transformations T.
But, in this linear algebra course, I want it to be a linear transformation. So here are
the rules for a linear transformation. Here, see, exactly, the two operations that we