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midterm-topics.html
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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Midterm Topics</title>
<meta name="author" content="Nathan Mull" />
<meta name="generator" content="Org Mode" />
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<body>
<div id="org-div-home-and-up"><a href="material.html">↩</a></div><div id="content" class="content">
<h1 class="title">Midterm Topics</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org2beb84e">Objects you should know</a></li>
<li><a href="#org7c83897">Things you should know how to do</a></li>
<li><a href="#org14a9d36">Facts you should know how to use</a></li>
<li><a href="#org32fbb21">Counterexample you should know</a></li>
</ul>
</div>
</div>
<p>
This a collection of things you should be comfortable with going into
the midterm. It's not exhaustive, and I can pretty much guarantee
that no question on the midterm will be a direct restatement of
anything below, but if you understand everything in this document, you
have the foundations to approach any question on the midterm.
</p>
<div id="outline-container-org2beb84e" class="outline-2">
<h2 id="org2beb84e">Objects you should know</h2>
<div class="outline-text-2" id="text-org2beb84e">
</div>
<ul class="org-ul">
<li><a id="org3fac5be"></a>Vectors<br />
<div class="outline-text-3" id="text-org3fac5be">
<ul class="org-ul">
<li>zero vector</li>
<li>standard basis vectors</li>
</ul>
</div>
</li>
<li><a id="orgdef0cf1"></a>Matrices<br />
<div class="outline-text-3" id="text-orgdef0cf1">
<ul class="org-ul">
<li>echelon forms</li>
<li>reduced echelon forms</li>
<li>zero matrix</li>
<li>identity matrix</li>
<li>2D linear transformation matrices for:
<ul class="org-ul">
<li>rotation</li>
<li>dilation</li>
<li>contraction</li>
<li>reflection</li>
<li>shearing</li>
<li>projection</li>
</ul></li>
<li>3D linear transformations for:
<ul class="org-ul">
<li>reflection</li>
<li>rotation</li>
</ul></li>
<li>matrix operations
<ul class="org-ul">
<li>multiplication</li>
<li>powers</li>
<li>inverse</li>
<li>transpose</li>
</ul></li>
</ul>
</div>
</li>
</ul>
</div>
<div id="outline-container-org7c83897" class="outline-2">
<h2 id="org7c83897">Things you should know how to do</h2>
<div class="outline-text-2" id="text-org7c83897">
</div>
<ul class="org-ul">
<li><a id="orgd305fa7"></a>Systems of linear equations<br />
<ul class="org-ul">
<li><a id="orgec27284"></a>Given a system of linear equations:<br />
<div class="outline-text-4" id="text-orgec27284">
<ul class="org-ul">
<li>determine if a given solution satisfies it</li>
<li>determine if it is consistent</li>
<li>find one of its solution, if one exists</li>
<li>find a general form solution which describes its solution set</li>
<li>write down its augmented matrix</li>
<li>write down its coefficient matrix</li>
<li>row reduce its augmented/coefficient matrix to echelon form</li>
<li>row reduce its augmented/coefficient matrix to reduced echelon form</li>
<li>determine if it has a unique solution</li>
<li>determine if it has infinitely many solutions</li>
</ul>
</div>
</li>
</ul>
</li>
<li><a id="orgf791a20"></a>Vectors<br />
<ul class="org-ul">
<li><a id="org9320252"></a>Given a vector equation:<br />
<div class="outline-text-4" id="text-org9320252">
<ul class="org-ul">
<li>determine if a given solution satsifies it</li>
<li>determine if it is consistent</li>
<li>find one of its solutions, if one exists</li>
<li>find a general form solution which describes its solution set</li>
<li>write down a system of linear equations which has the same solution set</li>
</ul>
</div>
</li>
<li><a id="orgb4238bb"></a>Given a set of vectors in ℝⁿ:<br />
<div class="outline-text-4" id="text-orgb4238bb">
<ul class="org-ul">
<li>compute a given linear combination of them</li>
<li>find a vector which lies in their span</li>
<li>determine if a given vector lies in their span</li>
<li>find a vector which does not lie in their span, if one exists</li>
<li>determine if set spans all of ℝⁿ</li>
<li>determine if the set is linearly independent</li>
<li>determine a dependence relation for them, if one exists</li>
<li>determine if its span is the same as the span of another given set of vectors</li>
</ul>
</div>
</li>
<li><a id="orgcb1b18b"></a>Given two vectors:<br />
<div class="outline-text-4" id="text-orgcb1b18b">
<ul class="org-ul">
<li>compute their inner product</li>
<li>find a linear equation which describes plane spanned by those vectors, given they are linearly independent</li>
</ul>
</div>
</li>
<li><a id="org4beba45"></a>Given one vector:<br />
<div class="outline-text-4" id="text-org4beba45">
<ul class="org-ul">
<li>find two plane equations whose intersection is its span, given it is nonzero</li>
<li>draw it, given it is in ℝ²</li>
</ul>
</div>
</li>
</ul>
</li>
<li><a id="org2bc4002"></a>Matrices<br />
<ul class="org-ul">
<li><a id="org1f41698"></a>Given a matrix equation:<br />
<div class="outline-text-4" id="text-org1f41698">
<ul class="org-ul">
<li>determine if a given vector is one of its solutions</li>
<li>determine if it is consistent</li>
<li>find one of its solutions, if one exists</li>
<li>write down a system of linear equations which has the same solution set</li>
<li>write down a vector equation which has the same solution set</li>
</ul>
</div>
</li>
<li><a id="org45d29ed"></a>Given a (m × n) matrix A:<br />
<div class="outline-text-4" id="text-org45d29ed">
<ul class="org-ul">
<li>determine if it is row equivalent to another given matrix</li>
<li>determine if it is in echelon form</li>
<li>determine if it is in reduced echelon form</li>
<li>determine its pivot positions</li>
<li>determine if the product Av for a given vector v is defined</li>
<li>compute the product Av for a given vector v, given it is defined</li>
<li>determine if the equation Ax = b has a solution for any choice of b</li>
<li>determine if the equation Ax = b has a unique solution for a given choice of b</li>
<li>determine if a given vector can be written as a linear combination of its columns</li>
<li>determine if its columns span ℝᵐ</li>
<li>determine if its columns are linearly independent</li>
</ul>
</div>
</li>
<li><a id="org056bd22"></a>Given a (2 × 2) matrix A:<br />
<div class="outline-text-4" id="text-org056bd22">
<ul class="org-ul">
<li>draw the effect on the transformation on the unit square</li>
</ul>
</div>
</li>
<li><a id="orgfaf5692"></a>Given a (n × n) matrix A:<br />
<div class="outline-text-4" id="text-orgfaf5692">
<ul class="org-ul">
<li>find its inverse</li>
<li>use the invertible matrix theorem to determine if it's invertible</li>
</ul>
</div>
</li>
</ul>
</li>
<li><a id="orge2cd2b4"></a>Linear transformations<br />
<ul class="org-ul">
<li><a id="orgddd81d8"></a>Given a transformation:<br />
<div class="outline-text-4" id="text-orgddd81d8">
<ul class="org-ul">
<li>identify its domain</li>
<li>identify its codomain</li>
<li>identify if its range is the same as its codomain</li>
<li>determine if it is linear</li>
</ul>
</div>
</li>
<li><a id="orgaa6882b"></a>Given a linear transformation:<br />
<div class="outline-text-4" id="text-orgaa6882b">
<ul class="org-ul">
<li>determine its value on a linear combination of vectors</li>
<li>find the matrix implementing a linear transformation given:
<ul class="org-ul">
<li>the input+output behavior of the transformation on a set of vectors</li>
<li>a geometric picture/description in ℝ² or ℝ³ of how the transformation behaves</li>
<li>an algebraic description of the linear transformation</li>
</ul></li>
<li>find a set of vectors which spans its range</li>
</ul>
</div>
</li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-org14a9d36" class="outline-2">
<h2 id="org14a9d36">Facts you should know how to use</h2>
<div class="outline-text-2" id="text-org14a9d36">
<ul class="org-ul">
<li>every matrix has a unique reduced echelon form</li>
<li>a matrix is the augmented matrix of a inconsistent system if and only if any of its echelon forms have a pivot in the last column</li>
<li>two vectors are linearly dependent if and only if they are co-linear</li>
<li>the following are equivalent for a (m × n) matrix A:
<ul class="org-ul">
<li>Ax = 0 has a unique solution</li>
<li>the columns of A are linearly independent</li>
<li>A has a pivot in every column</li>
</ul></li>
<li>the following are equivalent for a (m × n) matrix A:
<ul class="org-ul">
<li>Ax = b has a solution for any choice of b</li>
<li>the columns of A span ℝᵐ</li>
<li>A has a pivot in every row</li>
</ul></li>
<li>for a (m × n) matrix:
<ul class="org-ul">
<li>if m < n, then its columns are not linearly independent</li>
<li>if n < m, then its columns do not span ℝᵐ</li>
</ul></li>
<li>every condition of the invertible matrix theorem</li>
</ul>
</div>
</div>
<div id="outline-container-org32fbb21" class="outline-2">
<h2 id="org32fbb21">Counterexample you should know</h2>
<div class="outline-text-2" id="text-org32fbb21">
<ul class="org-ul">
<li>a consistent system of linear equations with more equations than unknowns</li>
<li>an inconsistent linear of linear equations with more unknowns than equations</li>
<li>two distinct echelon forms that are row+equivalent, but have the same entries in every pivot position</li>
<li>two different sized sets of vectors which have the same span</li>
<li>a set of linearly dependent vectors in which one cannot be written as a linear combination of other others</li>
<li>a set of linearly dependent vectors such that every proper subset is linear independent</li>
<li>a linear transformation which changes the length of some but not all vectors</li>
<li>a linear transformation which changes the direction of some but not all vectors</li>
</ul>
</div>
</div>
</div>
</body>
</html>