Review for Quiz 1

Lecture 9

Author

Affiliation

Minjae Park

Auburn University
MATH 2660 - Spring 2026

Published

January 28, 2026

Quiz information

Please bring an electronic device that can access WebAssign.
Be logged in and ready before 11:00 AM to avoid any technical issues.
The quiz is closed book. No materials are allowed, including the course website.
- You may bring blank scratch paper and a pen, or use an iPad/tablet for writing.
- If using a tablet, only a blank writing app (white paper) is allowed—do not open any other apps or materials.
Headphones are allowed if music helps you focus, but the volume must be low enough to hear announcements and not distract others. If it becomes disruptive, I will ask you to stop the music.

There will be about one question per topic, for a total of 8 questions.
You should aim to spend no more than 5 minutes per question.

A vector describes a direction together with a magnitude. It can be written as \[ \vec{v}=\langle a_1,a_2,\dots,a_n\rangle\in\mathbb{R}^n, \] or equivalently in column vector form.
A scalar is a real number representing a (signed) magnitude, denoted by \(c\in\mathbb{R}\).
Vectors can be added and scaled by real numbers to produce new vectors.
See the visualization for geometric intuition.

A linear combination of vectors \(\vec{v}_1,\dots,\vec{v}_k\) is a vector of the form \[ \vec{u} =c_1\vec{v}_1+\cdots+c_k\vec{v}_k =\sum_{i=1}^k c_i\vec{v}_i, \] where \(c_i\in\mathbb{R}\).
If we encode the vectors into a matrix \(A=[\vec{v}_1\ \cdots\ \vec{v}_k]\) and the coefficients into a vector \(\vec{c}=\langle c_1,\cdots,c_k \rangle\), then the linear combination can be written compactly as \[ \vec{u}=A\vec{c}. \]

A linear transformation \(A:\mathbb{R}^k\to\mathbb{R}^n\) maps a vector \(\vec{x}\in\mathbb{R}^k\) to \(A\vec{x}\in\mathbb{R}^n\) via multiplication by an \(n\times k\) matrix \(A\).
A linear transformation is completely determined by its action on the coordinate vectors \(\vec{e}_i\in\mathbb{R}^k\).
Each vector \(\vec{v}_i=A\vec{e}_i\) appears as the \(i\)-th column of \(A\).
In low dimensions—especially in the plane—linear transformations can often be visualized geometrically.
See the visualization for geometric intuition.

Let \[ A=[\vec{v}_1\ \cdots\ \vec{v}_k], \qquad C=[\vec{c}_1\ \cdots\ \vec{c}_m]. \]
The product \[ AC=[\vec{w}_1\ \cdots\ \vec{w}_m] \] is defined so that each column satisfies \(A\vec{c}_\ell=\vec{w}_\ell\).
Thus, matrix multiplication corresponds to applying a linear transformation to each column.
Be careful with dimensions: an \(n\times k\) matrix can be multiplied only to a \(k\times m\) matrix (from the left), producing an \(n\times m\) matrix.

A linear system can be written as \[ A\vec{x}=\vec{b}, \] where \(A\) is an \(n\times m\) matrix, \(\vec{x}\in\mathbb{R}^m\) is the vector of unknowns, and \(\vec{b}\in\mathbb{R}^n\) is given.
The corresponding augmented matrix is written as \([A\mid\vec{b}]\).
To solve a system, we apply Gauss–Jordan elimination using elementary row operations to obtain the reduced row-echelon form (RREF).
Depending on the RREF, the system may have a unique solution, infinitely many solutions, or no solution.

Let \(A:\mathbb{R}^n\to\mathbb{R}^n\) be a linear transformation, equivalently an \(n\times n\) matrix.
If another transformation \(B\) undoes what \(A\) does (and vice versa), then \[ AB=BA=I_n, \] where \(I_n\) is the identity matrix.
If such a matrix \(B\) exists, it is unique, and \(A\) is called invertible.
The matrix \(B\) is called the inverse of \(A\) and is denoted by \(A^{-1}\).
It can be computed using Gauss–Jordan elimination applied to the augmented matrix \([A\mid I_n]\).

Each elementary row operation can be interpreted as a linear transformation acting on a matrix.
An elementary matrix represents a single elementary row operation.
There are three types of elementary row operations:
- swapping two rows,
- scaling a row by a nonzero constant,
- adding a multiple of one row to another.

I will select several WebAssign preparation questions and work through them together.