Review for Quiz 2

Lecture 23

Author
Affiliation

Minjae Park

Auburn University
MATH 2660 - Spring 2026

Published

March 4, 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.

Topics

  1. Markov chains
  2. Abstract vector spaces
  3. Linear independence
  4. Spanning sets and bases
  5. Dot products (lengths, angles) and orthogonal projections
  6. Fundamental subspaces and dimensions (with RREF)
  7. Determinants and their properties

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.

Markov Chains

  • Given a context, construct the transition matrix \(P\).
  • \(P_{ij}\) = probability of moving from state \(j\) to state \(i\).
  • Each column must sum to 1: \(\sum_{i=1}^n p_{ij}=1\).
  • \(P^n\) represents the transition probabilities after \(n\) steps.

Example

A company has two customer states:
State 1 = Loyal, State 2 = Switcher.

Every month:

  • 20% of loyal customers switch.
  • 30% of switchers become loyal.

Construct the transition matrix \(P\) and verify that each column sums to 1.

Abstract Vector Spaces

  • To test whether a subset is a subspace, check:
    • Closed under addition
    • Closed under scalar multiplication
    • Contains the zero vector
  • The ambient space may consist of matrices, polynomials, or functions.

Examples

  1. In \(C(\mathbb{R})\), determine whether
    \(S=\{f \mid f(0)=0\}\) is a subspace.

  2. In \(P_2\), determine whether
    \(S=\{p(x) \mid p(1)=0\}\) is a subspace.

Linear Independence I

  • \(\vec{u}_1,\dots,\vec{u}_k\) are linearly independent if \[ c_1\vec{u}_1+\cdots+c_k\vec{u}_k=\vec{0} \] has only the trivial solution.
  • Form \(A=[\vec{u}_1\ \cdots\ \vec{u}_k]\) and solve \(A\vec{c}=\vec{0}\).

Examples

  1. Determine whether
    \(\vec{u}_1=\langle 1,0,1 \rangle\),
    \(\vec{u}_2=\langle 0,1,1 \rangle\),
    \(\vec{u}_3=\langle 1,1,2 \rangle\)
    are linearly independent.

  2. Given \(\vec{u}=\langle 1,2,0 \rangle\), \(\vec{v}=\langle 0,1,1 \rangle\), and \(\vec{w}=\langle 1,3,1 \rangle\), determine whether \(\vec{w}\) is in \(\mathrm{span}\{\vec{u},\vec{v}\}\).

Spanning Sets and Bases

  • A basis = spanning set + linear independence.
  • The number of basis vectors equals the dimension.

Example

Let
\(V=\mathrm{span}\{\langle 1,0,1 \rangle,\langle 0,1,1 \rangle,\langle 1,1,2 \rangle\}\).

Find the dimension of \(V\) and a basis for \(V\).

Linear Independence II

  • Pivot columns of \(\mathrm{RREF}(A)\) are linearly independent.
  • \(\mathrm{rank}(A)\) = number of independent columns.
  • If \(A\) is square: \(\det(A)\neq0\) ⇔ columns are independent.

Example

Let \[ A=\begin{pmatrix} 1&2&3\\ 2&4&6\\ 1&1&1 \end{pmatrix}. \]

  • Find \(\mathrm{rank}(A)\).
  • How many independent columns are there?
  • Find a basis for the column space \(C(A)\).

Dot Products and Projections

  • Length: \(\|\vec{v}\|=\sqrt{\vec{v}\cdot\vec{v}}\).
  • Angle: \[ \cos\theta=\frac{\vec{u}\cdot\vec{v}}{\|\vec{u}\|\|\vec{v}\|}. \]
  • Projection: \[ \mathrm{proj}_{\vec{u}}\vec{v} =\frac{\vec{v}\cdot\vec{u}}{\vec{u}\cdot\vec{u}}\vec{u}. \]

Example

Let \(\vec{u}=\langle 1,2,2 \rangle\) and \(\vec{v}=\langle 2,0,1 \rangle\).

  1. Compute \(\|\vec{u}\|\).
  2. Compute \(\cos\theta\).
  3. Determine whether the angle is acute, right, or obtuse.
  4. Compute \(\mathrm{proj}_{\vec{u}}\vec{v}\).

Fundamental Subspaces

  • From the RREF of \(A\):
    • Pivot columns → basis for column space
    • Free variables → basis for null space
  • \(\mathrm{rank}(A)\) = number of pivots.
  • Rank–nullity: \[ \mathrm{rank}(A)+\mathrm{nullity}(A)=\text{number of columns}. \] \[ \mathrm{rank}(A)+\mathrm{nullity}(A^T)=\text{number of rows}. \]

Example

Given the RREF \[ \begin{pmatrix} 1&0&2&0\\ 0&1&-1&0\\ 0&0&0&1 \end{pmatrix}, \]

  • Determine the rank and nullity.
  • Identify the pivot columns.
  • Find a basis for the null space \(N(A)\).

Example

Let \[ A= \begin{pmatrix} 1&0&2&1\\ 0&1&-1&3\\ 1&1&1&4\\ 0&1&2&7\\ 2&1&3&5 \end{pmatrix}. \]

Its RREF is \[ \mathrm{RREF}(A)= \begin{pmatrix} 1&0&2&0\\ 0&1&-1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}. \]

  1. Find \(\mathrm{rank}(A)\).
  2. Find \(\mathrm{nullity}(A)\).
  3. Find the left nullity of \(A\).
  4. Identify the pivot columns of \(A\).
  5. Find a basis for the column space \(C(A)\).
  6. Find a basis for the null space \(N(A)\).
  7. (*) Find a basis for the row space \(C(A^T)\).
  8. Let \(\vec{u}_i\) be the \(i\)th column of \(A\). Are \(\{\vec{u}_1,\vec{u}_2,\vec{u}_3\}\) linearly independent? Are \(\{\vec{u}_1,\vec{u}_3,\vec{u}_4\}\) linearly independent?

Determinants

  • \(2\times2\) formula: \[ \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc. \]
  • The determinants of elementary matrices are easy to compute.
  • Key properties:
    • \(\det(AB)=\det(A)\det(B)\)
    • \(\det(A^{-1})=1/\det(A)\)
    • \(\det(cA) = c^n A\) (You apply row scaling \(n\) times for each row.)
  • \(\det(A)\neq0\) ⇔ invertible ⇔ full rank.

Examples

  1. Compute \[ \det\begin{pmatrix}3&1\\2&5\end{pmatrix}. \]

  2. Compute the determinants of upper traingular matrices.

  3. Let \(A, B\) be two \(3\times 3\) matrices. Suppose \(\det(A)=2\) and \(\det(B)=-3\).
    Compute \(\det(ABA)\), \(\det(A^{-1})\), and \(\det(-3A)\).