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. The activities in this section are not strictly necessary to work through the rest of this text.
Activity 1.1.2.
Consider the augmented 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 necessary 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 necessary and sufficient conditions for the columns to be linearly independent.
(b)
Describe necessary and sufficient conditions for the columns to be a spanning set.
(c)
Describe necessary and sufficient conditions for the columns to be a basis for \(\mathbb{R}^m\text{.}\)
Here, we recall the geometric interpretation of the determinant.
Observation 1.1.4.
Recall that each \(n\times m\) matrix may be thought of as a linear transformation from \(\mathbb{R}^m\to \mathbb{R}^n\text{.}\) When \(m=n\) we may define the determinant of the transformation. The determinant has numerous algebraic properties one learns about in a linear algebra course, but it also has a geometric interpretation. Roughly speaking, the determinant measures how the transformation changes the unit \(n\) dimensional cube, with the magnitude of the determinant measuring the \(n\)-dimensional volume of the cube after transformation, and the sign measuring whether or not the orientation of the cube is preserved or reversed.
Some resources for linear algebra define the determinant algebraically, then prove that it has special geometric properties. In many ways this is a natural approach to introduce the subject to students 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 1.1.5.
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( \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] \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( \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] \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( \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] \right)\\
=\amp \det \left( \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] \right)
\end{align*}
geometrically.
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?
