Activity 1.1.1.
Let
\begin{equation*}
A=\begin{bmatrix}1 & a_2 & a_3 & \cdots & a_n \end{bmatrix}.
\end{equation*}
What is \(\dim(\operatorname{null}(A))\text{?}\)
Section 1.1 A Brief Geometric Review of Linear Algebra
In this introductory section, we do not begin linear optimization. Instead, we recall a few concepts from Linear Algebra, and examine them through a geometric lense, setting the stage for our mindset going forward.
Activity 1.1.2.
Consider the augemnted matrix
\begin{equation*}
M=\left[\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\
a_{21} & a_{22} & \cdots & a_{2n}& b_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\
\end{array}\right] = (A|b).
\end{equation*}
(a)
Given a fixed \(n\text{,}\) what is a neccesary condition for the values of \(m\) so that the system of equations encoded by \(M\) has a unique solution?
(b)
What does this mean geometrically?
(c)
If the rows of \(A\) are independent, and \(m\lt n\text{,}\) then what is the dimension of the solution space of \(M\text{?}\)
Activity 1.1.3.
Consider the matrix
\begin{equation*}
M=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}\right].
\end{equation*}
(a)
Describe neccesary and sufficient conditions for the columns to be linearly independent.
(b)
Describe neccesary and sufficient conditions for the columns to be a spanning set.
(c)
Describe neccesary and sufficient conditions for the columns to be a basis for \(\mathbb{R}^m\text{.}\)
Activity 1.1.4.
Consider the square matrix
\begin{equation*}
A=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{array}\right].
\end{equation*}
(a)
Explain why for any constant \(c\) and \(1\leq j\leq n-1\)
\begin{equation*}
\det(A)=\det \left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1}+ca_{j1} & a_{n2}+ca_{j2} & \cdots & a_{nn} +ca_{jn} \\
\end{array}\right]
\end{equation*}
geometrically, i.e. without cofactor expansion.
(b)
Explain why for any constant \(c\) and \(1\leq j\leq n\)
\begin{equation*}
\det \left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
ca_{n1} & ca_{n2} & \cdots & ca_{nn} \\
\end{array}\right] = c\det(A)
\end{equation*}
geometrically.
(c)
Explain why for any constant \(c\) and \(1\leq j\leq n\)
\begin{align*}
\amp \det \left[\begin{array}{ccccc}
a_{11} & a_{12} & \cdots & a_{1(n-1)} & 0 \\
a_{21} & a_{22} & \cdots & a_{2(n-1)} & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{(n-1)1} & a_{(n-1)2} & \cdots & a_{(n-1)(n-1)} & 0 \\
0 & 0 & \cdots & 0 & 1 \\
\end{array}\right]\\
=\amp \det\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1(n-1)} \\
a_{21} & a_{22} & \cdots & a_{2(n-1)} \\
\vdots & \vdots & \ddots & \vdots \\
a_{(n-1)1} & a_{(n-1)2} & \cdots & a_{(n-1)(n-1)} \\
\end{array}\right]
\end{align*}
geometrically.
Observation 1.1.5.
Some resource for linear algebra define the determinant algebraically, then prove that it has these special geometric properties. In many ways this is a natural approach to introduce the subject to student whose background is primarily algebraic. However, in my opinion, this is backwards. It makes far more sense to approach the determinant geometrically first, there is a property of transformations we want to measure, we call this quantity the determinant, it happens to have cool algebraic properties.
Activity 1.1.6.
Michael Atiyah (1929 - 2019), mathematician and Field’s medalist (1966), once said
“Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.””
What do you suppose Dr. Atiyah meant by this quote? What does it mean to you? How might this sentiment have impacted your mathematical journey or education?
