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Continue **reading** Section 3.5. We aren't covering 3.4.
Work through recommended homework questions.

Five **practice midterms** have been posted on the course web page.

**Next office hour:** Monday, 3:00-3:30, MC103B.

**Help Centers:** Monday-Friday 2:30-6:30 in MC 106, but not during
reading week.

After today, we are halfway done the course!

**Definition:** An **inverse** of an $n \times n$ matrix $A$ is an $n \times n$
matrix $A'$ such that
$$
A A' = I \qtext{and} A' A = I .
$$
If such an $A'$ exists, we say that $A$ is **invertible**.

**Theorem 3.6:** If $A$ is an invertible matrix, then its inverse is unique.

We write $A^{-1}$ for **the** inverse of $A$, when $A$ is invertible.

**Theorem 3.8:** The matrix $A = \bmat{cc} a & b \\ c & d \emat$ is
invertible if and only if $ad - bc \neq 0$. When this is the case,
$$
A^{-1} = \frac{1}{ad-bc} \, \bmat{rr} \red{d} & \red{-}b \\ \red{-}c & \red{a} \emat .
$$

We call $ad-bc$ the **determinant** of $A$, and write it $\det A$.

- $A^{-1}$ is invertible and $(A^{-1})^{-1} = {A}$
- If $c$ is a non-zero scalar, then $cA$ is invertible and $(cA)^{-1} = {\frac{1}{c} A^{-1}}$
- $AB$ is invertible and $(AB)^{-1} = {B^{-1} A^{-1}}$ (socks and shoes rule)
- $A^T$ is invertible and $(A^T)^{-1} = {(A^{-1})^T}$
- $A^n$ is invertible for all $n \geq 0$ and $(A^n)^{-1} = {(A^{-1})^n}$

**Remark:** There is no formula for $(A+B)^{-1}$.
In fact, $A+B$ might not be invertible, even if $A$ and $B$ are.

**Theorem 3.12:**
Let $A$ be an $n \times n$ matrix. The following are equivalent:

a. $A$ is invertible.

b. $A \vx = \vb$ has a unique solution for every $\vb \in \R^n$.

c. $A \vx = \vec 0$ has only the trivial (zero) solution.

d. The reduced row echelon form of $A$ is $I_n$.

**Theorem 3.13:** Let $A$ be a square matrix. If $B$ is a square
matrix such that either $AB=I$ or $BA=I$, then $A$ is invertible
and $B = A^{-1}$.

**Theorem 3.14**: Let $A$ be a square matrix. If a sequence of row
operations reduces $A$ to $I$, then the **same** sequence of row
operations transforms $I$ into $A^{-1}$.

This gives a general purpose method for determining whether a matrix $A$ is invertible, and finding the inverse:

1. Form the $n \times 2n$ matrix $[A \mid I\,]$.

2. Use row operations to get it into reduced row echelon form.

3. If a zero row appears in the left-hand portion, then $A$ is not invertible.

4. Otherwise, $A$ will turn into $I$, and the right hand portion is $A^{-1}$.

**Definition:** A **subspace** of $\R^n$ is any collection $S$ of
vectors in $\R^n$ such that:

1. The zero vector $\vec 0$ is in $S$.

2. $S$ is **closed under addition**:
If $\vu$ and $\vv$ are in $S$, then $\vu + \vv$ is in $S$.

3. $S$ is **closed under scalar multiplication**:
If $\vu$ is in $S$ and $c$ is any scalar, then $c \vu$ is in $S$.

Conditions (2) and (3) together are the same as saying that $S$ is
**closed under linear combinations**.

**Example:** $\R^n$ is a subspace of $\R^n$.
Also, $S = \{ \vec 0 \}$ is a subspace of $\R^n$.

**Example:** A plane $\cP$ through the origin in $\R^3$ is a subspace.
Applet.

Here's an algebraic argument.
Suppose $\vv_1$ and $\vv_2$ are direction vectors for $\cP$,
so $\cP = \span(\vv_1, \vv_2)$.

(1) $\vec 0$ is in $\cP$, since $\vec 0 = 0 \vv_1 + 0 \vv_2$.

(2) If $\vu = c_1 \vv_1 + c_2 \vv_2$ and $\vv = d_1 \vv_1 + d_2 \vv_2$,
then
$$
\begin{aligned}
\vu + \vv\ &= (c_1 \vv_1 + c_2 \vv_2) + (d_1 \vv_1 + d_2 \vv_2) \\
&= (c_1 + d_1) \vv_1 + (c_2 + d_2) \vv_2
\end{aligned}
$$
which is in $\span(\vv_1, \vv_2)$ as well.

(3) For any scalar $c$,
$$
c \vu = c (c_1 \vv_1 + c_2 \vv_2) = (c c_1) \vv_1 + (c c_2) \vv_2
$$
which is also in $\span(\vv_1, \vv_2)$.

On the other hand, a plane **not** through the origin is not a subspace.
It of course fails (1), but the other conditions fail as well, as shown in
the applet.

As another example, a line through the origin in $\R^3$ is also a subspace.

The **same** method as used above proves:

**Theorem 3.19:** Let $\vv_1, \vv_2, \ldots, \vv_k$ be vectors in $\R^n$.
Then $\span(\vv_1, \ldots, \vv_k)$ is a subspace of $\R^n$.

See text. We call $\span(\vv_1, \ldots, \vv_k)$ the **subspace spanned
by $\vv_1, \ldots, \vv_k$.**
This generalizes the idea of a line or a plane through the origin.

**Example:**
Is the set of vectors $\colll x y z$ with $x = y + z$
a subspace of $\R^3$?

See Example 3.38 in the text for a similar question.

**Example:**
Is the set of vectors $\ccolll x y z$ with $x = y + z + 1$
a subspace of $\R^3$?

**Example:**
Is the set of vectors $\ccoll x y $ with $y = \sin(x)$
a subspace of $\R^2$?

**Theorem 3.21:** Let $A$ be an $m \times n$ matrix and let $N$ be the
set of solutions of the homogeneous system $A \vx = \vec 0$.
Then $N$ is a subspace of $\R^n$.

**Proof:**
(1) Since $A \, \vec 0_n = \vec 0_m$, the zero vector $\vec 0_n$ is in $N$.

(2) Let $\vu$ and $\vv$ be in $N$, so $A \vu = \vec 0$ and $A \vv = \vec 0$.
Then
$$ A (\vu + \vv) = A \vu + A \vv = \vec 0 + \vec 0 = \vec 0 $$
so $\vu + \vv$ is in $N$.

(3) If $c$ is a scalar and $\vu$ is in $N$, then
$$ A (c \vu) = c A \vu = c \, \vec 0 = \vec 0 $$
so $c \vu$ is in $N$. $\qquad \Box$

**Aside:** At this point, the book states **Theorem 3.22**, which
says that every linear system has no solution, one solution or infinitely
many solutions, and gives a proof of this. We already know this is true,
using Theorem 2.2 from Section 2.2 (see Lecture 9).
The proof given here is in a sense better, since it doesn't rely on
knowing anything about row echelon form, but I won't use class time
to cover it.

Spans and null spaces are the *two main* sources of subspaces.

**Definition:** Let $A$ be an $m \times n$ matrix.

1. The **row space** of $A$ is the subspace $\row(A)$ of $\R^n$ spanned
by the rows of $A$.

2. The **column space** of $A$ is the subspace $\col(A)$ of $\R^m$ spanned
by the columns of $A$.

3. The **null space** of $A$ is the subspace $\null(A)$ of $\R^n$
consisting of the solutions to the system $A \vx = \vec 0$.

**Example:** The column space of $A = \bmat{rr} 1 & 2 \\ 3 & 4 \emat$
is $\span(\coll 1 3, \coll 2 4)$.
A vector $\vb$ is a linear combination of these columns if and only if
the system $A \vx = \vb$ has a solution.
But since $A$ is invertible (its determinant is $4 - 6 = -2 \neq 0$),
every such system has a (unique) solution.
So $\col(A) = \R^2$.

The row space of $A$ is the same as the column space of $A^T$, so by a similar argument, this is all of $\R^2$ as well.

The null space of $A$ consists of the vectors $\coll x y$ such that $A \coll x y = \vec 0$. That is, \[ x \coll 1 3 + y \coll 2 4 = \vec 0 . \] Since those columns are linearly independent, $\null(A) = \{ \vec 0 \}$.

**Example:** The column space of $A = \bmat{rr} 1 & 2 \\ 3 & 4 \\ 5 & 6 \emat$
is the span of the two columns, which is a subspace of $\R^3$.
Since the columns are linearly independent, this is a plane through the origin in $\R^3$.

Determine whether $\colll 2 0 1$ and $\colll 2 0 {-2}$ are in $\col(A)$. (On board.)

The row space of $A$ is the span of the three rows. But we already saw that the span of the first two rows is $\R^2$, so the span of all three rows is also $\R^2$. So $\row(A) = \R^2$.

Again, since the columns are linearly independent, $\null(A) = \{ \vec 0 \}$.

**Example:** Find the null space of $A = \bmat{rr} 1 & 2 \\ -2 & -4 \emat$.

We want to solve the system $A \vx = \vec 0$, so we row reduce $\bmat{rr|r} 1 & 2 & 0 \\ -2 & -4 & 0 \emat$ to $\bmat{rr|r} 1 & 2 & 0 \\ 0 & 0 & 0 \emat$. Then $y = t$ and $x = -2t$ are the solutions, so \[ \null(A) = \left\{ \coll {-2t} t \right\} . \]

Next we will explain the best way to describe a subspace.

**Definition:** A **basis** for a subspace $S$ of $\R^n$ is a
set of vectors $\vv_1, \ldots, \vv_k$ such that:

1. $S = \span(\vv_1, \ldots, \vv_k)$, and

2. $\vv_1, \ldots, \vv_k$ are linearly independent.

Condition (2) ensures that none of the vectors is redundant, so we aren't being wasteful. Giving a basis for a subspace is a good way to "describe" it.

**Example 3.42:** The standard unit vectors $\ve_1, \ldots, \ve_n$
in $\R^n$ are linearly independent and span $\R^n$, so they form a
basis of $\R^n$ called the **standard basis**.

**Example:**
We saw above that $\coll 1 3$ and $\coll 2 4$ span $\R^2$.
They are also linearly independent, so they are a basis for $\R^2$.

Note that $\coll 1 0$ and $\coll 0 1$ are another basis for $\R^2$. A subspace will in general have many bases, but we'll see soon that they all have the same number of vectors! (Grammar: one basis, two bases.)

Next class we will continue talking about bases and will discuss systematic methods for finding the three subspaces associated to a matrix $A$.