Activity 1
- Consider the matrix
$$
M_1 =
\begin{bmatrix}
3 & 0 & 2 & 5\\
-1 & 3 & -2 & 3\\
4 & 9 & 0 & -5\\
\end{bmatrix}
$$
- What is the dimension of this matrix?
- What are the row vectors of this matrix?
- What are the column vectors of his matrix?
-
Suppose we want to define the meaning of
"matrix-vector product" $M\cdot \boldsymbol{x}$, for
matrix $M$ and column vector
$\boldsymbol{x}$,
so that it is equivalent to
- the linear combination of the
column vectors of $M$ where the coefficients are the
components of $\boldsymbol{x}$, or
- the linear transformation defined by row vectors of
$M$, applied to the vector $\boldsymbol{x}$.
... hopefully you realize that both of these are in fact
equivalent.
-
Looking at $M_1$ in the above example,
what dimension must the column vector $\boldsymbol{x}$
have in order to have the linear combination of the
columns (or dot products defining the linear
transformation) make sense?
-
In general, if $M$ is an $m\times n$ matrix, in order
for the product $M \cdot \boldsymbol{x}$
make sense, what dimension must the column vector $\boldsymbol{x}$
have?
-
Assuming $M$ and $\boldsymbol{x}$ have compatible
dimensions, how should result of the matrix vector product
$M \cdot \boldsymbol{x}$ be defined?
Activity 2
-
Compute the following matrix products:
$$
\begin{array}{ccccc}
\begin{bmatrix}
4&0&-2\\
1&-3&4
\end{bmatrix}
\cdot
\begin{bmatrix}
2\\
-1\\
1/2
\end{bmatrix}
= \ \ \ \ \ &,&
\begin{bmatrix}
2&3\\
-1&0\\
5&2
\end{bmatrix}
\cdot
\begin{bmatrix}
2\\
1/2
\end{bmatrix}
= \ \ \ \ \ &,&
\begin{bmatrix}
3&-2\\
5&4
\end{bmatrix}
\cdot
\begin{bmatrix}
-7\\
4
\end{bmatrix}
\end{array}
=
$$
-
Do the following two multiplications:
$$
\begin{array}{cccc}
%%%%%
\begin{bmatrix}
3&-4\\
2&5\\
1&-1
\end{bmatrix}
\cdot
\begin{bmatrix}
1\\
0
\end{bmatrix}
= \ \ \ \ \ &,&
%%%%%
\begin{bmatrix}
3&-4\\
2&5\\
1&-1
\end{bmatrix}
\cdot
\begin{bmatrix}
0\\
1
\end{bmatrix}
=
%%%%%
\end{array}
$$
-
Make a hypothesis: Let $M$ by an $m\times n$ matrix and
$\boldsymbol{x}$ be an $n$-dimensional column vector in
which the $i$th component is $1$, and all other components
are $0$. Then $M\cdot \boldsymbol{x} = $ ....
-
Find a $2\times 2$ matrix with the property that when you
multiply it and any 2-dimensional vector $\boldsymbol{x}$,
you get that same $\boldsymbol{x}$ back again. I.e.
$$
\begin{bmatrix}
?&?\\
?&?
\end{bmatrix}
\cdot
\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
=
\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
$$