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# Multiplying a matrix by a column vector

Video transcript

So we're multiplying
matrix A here by vector w. And they ask us
what is A times w? So let me get my
scratch pad out. And let's think about this. So we have matrix
A times vector w. So just as a reminder,
Aw-- and I'll write it in bold
right over here-- Aw. And I could write it either
as a vector like that, or I could bold that as well. Aw. It's lowercase for a vector,
uppercase for a matrix. This is the same thing
as matrix A times vector w-- let me do that same color--
which is the same thing. This is matrix A. So it's,
A is 0, 3, 5, 5, 5, 2 times vector w. W is the vector 3, 4, 3. So we have a matrix with
two rows and three columns, so it is a 2 by 3
matrix, times a vector. And this is a column vector. It is three rows in one column. So you could view this
as a 3 by 1 matrix. And so, deciding whether this
is even a valid operation, it's just like
multiplying two matrices. You can view this column vector
is really just a 3 by 1 matrix. And the only way that matrix
multiplication is defined is if the columns
in this matrix is equal to the rows
in this matrix. And we see that is
the case, this vector. This matrix, matrix A, has
three columns: 1, 2, 3. And this and vector w
has exactly three rows. So matrix multiplication,
or matrix times vector multiplication,
is defined here. And the way that
we're going to do it, we're going to end
up with a 2 by 1. You could call it
a 2 by 1 matrix. It's going to have two
rows in one column. Or you could view it as
a column vector, when you have two rows
and one column. But either way,
let's actually think about how we would compute it. So it's going to have
two rows and one column. Then I'm going
give a lot of space so that we can actually
do the calculation. So the top entry
right over here, we're going to get
the row information from our first matrix, and
then the column information. There's only one column here. But we're going to get that
from this matrix or this vector, whatever you want to call it. So we're going to have 0 times
3, plus 3 times 4, plus 5 times 3. Now the second entry
right over here is going to be the
second row here. Essentially, the dot product
of that with this vector. And if you don't know
what the dot product is, I'm essentially
about to do that. It's going to be-- you take
each corresponding term, take their product, and then
add up everything together. So you have 5 times 3, plus
5 times 4, plus 2 times 3. And this is going to
simplify to-- This top term, this is a 0. This is 12 plus 15, which is 27. And then the second term,
this is 15 plus 20 plus 6. So this is 35 plus 6 is 41. So it's a column vector,
two rows, 27 and 41. Now let's input that, 27 and 41. So we get 27. And here we can put 41,
and check our answer, and we got it right.