Let us begin by introducing basic notation for matrices and vectors.

We'll use to denote the set of real-numbers and to denote the set of complex numbers.

We write the space of all real-valued matrices as
. Each
Sometimes, I'll write:
instead.
With only a few exceptions, matrices are written as *bold, capital*
letters. Sometimes, we'll use a capital greek letter.
Matrix elements are written as sub-scripted, *unbold*
letters.

When clear from context,
instead, e.g. instead of .

**In class** I'll usually write matrices with just upper-case letters.
If you are unsure if something
is a matrix or an element, raise your hand and ask, or *quietly*
ask a neighbor.

Another notation for is

Sometimes this is nicer to write on the
board.

We write the set of length- real-valued vectors as .
Each
Vectors are denoted by *lowercase, bold* letters.
As with matrices, elements are sub-scripted, *unbold* letters.
Sometimes, we'll write vector elements as
Usually, this choice is motivated by a particular application.
*Throughout the class, vectors are column vectors.*

**In class** I'll usually write vectors with just lower-case letters and
*will try* to follow the convention of underlining them.

Lower-case greek letters are scalars.

Identify the following:

**Transpose** Let , then

*Example*

**Hermitian** (Also called conjugate transpose)
Let , then

*Example*

**Addition**
Let and , then

*Example*
.

**Scalar Multiplication**
Let and , then

*Example*

**Matrix Multiplication**
Let and , then

**Matrix-vector Multiplication**
Let and , then
This operation is just a special case of matrix multiplication
that follows from treating and as and
matrices, respectively.

**Vector addition**, **Scalar vector multiplication**
These are just special cases of matrix addition and
scalar matrix multiplication where vectors are viewed
as matrices.

It is often useful to represent a matrix as a collection of vectors.
In this case, we write
where each . *This form corresponds to a
partition into columns.*

Alternatively, we may wish to partition a matrix into rows. Here, each .

Using the column partitioning: And with the row partitioning:

Another useful partitioned representation of a matrix is into
blocks:
or

Here, the sizes "just have to work out" in the vernacular.
Formally, all must have the same number
of rows and all must have the same number of columns.