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<title>Linear Algebra Lecture 1 Notes</title>
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<!-- ===== LECTURE TITLE ===== -->
<h1>Linear Algebra Lecture 1</h1>
<p style="text-align: center; color: #666; font-style: italic;">Basic Algebra of Vector Spaces</p>
<!-- ===== CONTENT BOXES ===== -->
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('3:02', 'Course Introduction', 'default', this)">
<span class="play-icon">▶</span>3:02
</div>
<h3>Course Overview</h3>
<p>This course revisits familiar concepts from first year vectors and matrices, but with a very different approach. The emphasis shifts from concrete objects (column vectors, arrays of numbers) to the abstract algebraic perspective that pure mathematicians use.</p>
<p>We'll develop the language of abstract algebra to discuss these concepts, preparing for more advanced courses like representation theory and algebraic geometry. The course is also highly applicable, particularly for infinite dimensional vector spaces and function spaces.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('6:40', 'Vector Space Definition', 'definition', this)">
<span class="play-icon">▶</span>6:40
</div>
<h3>Vector Space</h3>
<p>An <strong>F-vector space</strong> (or vector space over F) consists of:</p>
<ol>
<li>An abelian group $(V, +)$ with underlying set $V$ and operation written as addition</li>
<li>The identity element written as $\underline{0}$ (zero vector)</li>
<li>A function $F \times V \to V$ called <strong>scalar multiplication</strong>, written as $(\lambda, v) \mapsto \lambda v$ or $\lambda \cdot v$</li>
</ol>
<p>These operations satisfy the following axioms for all $v, w \in V$ and $\lambda, \mu \in F$:</p>
<ol>
<li>$(\lambda + \mu) \cdot v = \lambda \cdot v + \mu \cdot v$ (distributivity over field addition)</li>
<li>$\lambda \cdot (v + w) = \lambda \cdot v + \lambda \cdot w$ (distributivity over vector addition)</li>
<li>$(\lambda \mu) \cdot v = \lambda \cdot (\mu \cdot v)$ (associativity of scalar multiplication)</li>
<li>$1 \cdot v = v$ (multiplicative identity acts as identity)</li>
</ol>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('11:33', 'Notation and Terminology', 'default', this)">
<span class="play-icon">▶</span>11:33
</div>
<h3>Notation and Terminology</h3>
<p><strong>Elements of V</strong> are called <strong>vectors</strong> and are not underlined (except the zero vector $\underline{0}$).</p>
<p><strong>Elements of F</strong> are called <strong>scalars</strong>.</p>
<p>The zero vector $\underline{0}$ is the additive identity of the group $(V, +)$ and is distinct from the element $0$ in the field $F$.</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('12:47', 'Column Vectors Example', 'example', this)">
<span class="play-icon">▶</span>12:47
</div>
<h3>Column Vectors</h3>
<p>The most familiar example is $F^n$, the space of $n$-dimensional column vectors with entries in the field $F$:</p>
<ul>
<li>$\mathbb{R}^n$ is the space of real $n$-dimensional column vectors</li>
<li>$\mathbb{C}^n$ is the space of complex $n$-dimensional column vectors</li>
</ul>
<p>Addition is defined component-wise, and scalar multiplication is defined element-wise.</p>
<p><strong>Important observation:</strong> $\mathbb{C}^n$ is both a complex vector space and a real vector space. We can restrict scalar multiplication to only real numbers, making it a real vector space as well.</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('15:33', 'Matrices Example', 'example', this)">
<span class="play-icon">▶</span>15:33
</div>
<h3>Matrices</h3>
<p>The space $M_{m,n}(F)$ of $m \times n$ matrices over the field $F$ forms a vector space. This is essentially the same as $F^{mn}$ (column vectors of length $mn$), just arranged in a rectangular format rather than a column.</p>
<p>Matrices have additional structure beyond their vector space structure (matrix multiplication, etc.).</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('17:15', 'Function Spaces Example', 'example', this)">
<span class="play-icon">▶</span>17:15
</div>
<h3>Function Spaces</h3>
<p>For any non-empty set $X$, the space $F^X$ of functions from $X$ to $F$ is a vector space with operations defined pointwise:</p>
<ul>
<li>$(f + g)(x) = f(x) + g(x)$ for all $x \in X$</li>
<li>$(\lambda f)(x) = \lambda \cdot f(x)$ for all $x \in X$</li>
</ul>
<p>This includes important subspaces like:</p>
<ul>
<li>$C(\mathbb{R})$ - continuous real-valued functions</li>
<li>$C^\infty(\mathbb{R})$ - infinitely differentiable functions</li>
<li>$P(\mathbb{R})$ - polynomial functions</li>
</ul>
</div>
<div class="claim content-box">
<div class="timestamp" onclick="openVideoSide('19:17', 'Additional Properties', 'claim', this)">
<span class="play-icon">▶</span>19:17
</div>
<h3>Additional Properties</h3>
<p>Other familiar properties of vector spaces follow from the axioms:</p>
<ul>
<li>$0 \cdot v = \underline{0}$ (scalar multiplication by zero gives zero vector)</li>
<li>$(-1) \cdot v = -v$ (scalar multiplication by -1 gives additive inverse)</li>
</ul>
<p><em>These can be verified as exercises using the given axioms.</em></p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('20:54', 'Subspace Definition', 'definition', this)">
<span class="play-icon">▶</span>20:54
</div>
<h3>Subspace</h3>
<p>A <strong>subspace</strong> $U$ of an $F$-vector space $V$ is a subset $U \subseteq V$ that is itself an $F$-vector space under the same operations as $V$.</p>
<p><strong>Equivalent condition:</strong> $U$ is a subspace if and only if:</p>
<ol>
<li>$(U, +)$ is a subgroup of $(V, +)$</li>
<li>Scalar multiplication preserves $U$: for all $\lambda \in F$ and $u \in U$, we have $\lambda u \in U$</li>
</ol>
</div>
<div class="theorem content-box">
<div class="timestamp" onclick="openVideoSide('24:45', 'Subspace Test', 'theorem', this)">
<span class="play-icon">▶</span>24:45
</div>
<h3>Subspace Test</h3>
<p>A subset $U \subseteq V$ is a subspace if and only if:</p>
<ol>
<li>$U \neq \emptyset$ (non-empty)</li>
<li>For all $u, w \in U$ and $\lambda \in F$: $u + \lambda w \in U$</li>
</ol>
</div>
<div class="proof content-box">
<div class="timestamp" onclick="openVideoSide('26:27', 'Subspace Test Proof', 'proof', this)">
<span class="play-icon">▶</span>26:27
</div>
<h3>Proof of Subspace Test</h3>
<p><strong>Forward direction:</strong> If $U$ is a subspace, then it's non-empty (contains $\underline{0}$) and closed under addition and scalar multiplication, so $u + \lambda w \in U$.</p>
<p><strong>Reverse direction:</strong> Suppose $U$ satisfies both conditions. Taking $\lambda = -1$ in condition 2 gives $u - w \in U$ for all $u, w \in U$. Combined with non-emptiness, this shows $U$ is a subgroup by the subgroup test from group theory.</p>
<p>Taking $u = \underline{0}$ in condition 2 gives $\lambda w \in U$ for all $\lambda \in F$ and $w \in U$, so scalar multiplication preserves $U$. Therefore $U$ is a subspace.</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('31:22', 'Subspace Examples', 'example', this)">
<span class="play-icon">▶</span>31:22
</div>
<h3>Subspace Examples</h3>
<p><strong>Example 1:</strong> The set $\{(x,y,z) \in \mathbb{R}^3 : x + y + z = t\}$ is a subspace of $\mathbb{R}^3$ if and only if $t = 0$.</p>
<p><strong>Example 2:</strong> In the vector space of all functions $\mathbb{R} \to \mathbb{R}$:</p>
<ul>
<li>$C(\mathbb{R})$ (continuous functions) is a subspace</li>
<li>$C^\infty(\mathbb{R})$ (infinitely differentiable functions) is a subspace</li>
<li>$P(\mathbb{R})$ (polynomial functions) is a subspace</li>
</ul>
<p>Each inclusion $P(\mathbb{R}) \subseteq C^\infty(\mathbb{R}) \subseteq C(\mathbb{R}) \subseteq \mathbb{R}^{\mathbb{R}}$ is a subspace inclusion.</p>
</div>
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<div class="timestamp" onclick="openVideoSide('36:43', 'Intersection of Subspaces', 'theorem', this)">
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<h3>Intersection of Subspaces</h3>
<p>If $U$ and $W$ are subspaces of $V$, then $U \cap W$ is also a subspace of $V$.</p>
</div>
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<div class="timestamp" onclick="openVideoSide('37:16', 'Intersection Proof', 'proof', this)">
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<h3>Proof of Intersection Property</h3>
<p><strong>Non-empty:</strong> Both $U$ and $W$ contain $\underline{0}$, so $\underline{0} \in U \cap W$.</p>
<p><strong>Subspace test:</strong> For $x, y \in U \cap W$ and $\lambda \in F$:</p>
<ul>
<li>Since $U$ is a subspace: $x + \lambda y \in U$</li>
<li>Since $W$ is a subspace: $x + \lambda y \in W$</li>
<li>Therefore: $x + \lambda y \in U \cap W$</li>
</ul>
<p>Hence $U \cap W$ satisfies the subspace test.</p>
</div>
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<div class="timestamp" onclick="openVideoSide('39:57', 'Union Warning', 'default', this)">
<span class="play-icon">▶</span>39:57
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<h3>Important Warning: Unions</h3>
<p><strong>The union of subspaces is almost never a subspace.</strong></p>
<p>This is a common mistake. If you want a subspace that contains both $U$ and $W$, you should use the <strong>subspace sum</strong> instead.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('41:22', 'Subspace Sum Definition', 'definition', this)">
<span class="play-icon">▶</span>41:22
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<h3>Subspace Sum</h3>
<p>For subspaces $U$ and $W$ of $V$, the <strong>subspace sum</strong> is defined as:</p>
<p>$$U + W = \{u + w : u \in U, w \in W\}$$</p>
<p>This is the smallest subspace of $V$ containing both $U$ and $W$.</p>
</div>
<div class="theorem content-box">
<div class="timestamp" onclick="openVideoSide('42:09', 'Subspace Sum Properties', 'theorem', this)">
<span class="play-icon">▶</span>42:09
</div>
<h3>Subspace Sum Properties</h3>
<p>For any subspaces $U$ and $W$ of $V$:</p>
<ol>
<li>$U + W$ is always a subspace of $V$</li>
<li>$U + W$ contains both $U$ and $W$</li>
<li>$U + W$ is the smallest subspace containing both $U$ and $W$</li>
</ol>
<p><em>Proof by subspace test (exercise).</em></p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('45:22', 'Linear Map Definition', 'definition', this)">
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<h3>Linear Map</h3>
<p>A <strong>linear map</strong> (or linear transformation) from $V$ to $W$ is a function $\alpha: V \to W$ such that for all $u, v \in V$ and $\lambda \in F$:</p>
<ol>
<li>$\alpha(u + v) = \alpha(u) + \alpha(v)$ (preserves addition)</li>
<li>$\alpha(\lambda v) = \lambda \alpha(v)$ (preserves scalar multiplication)</li>
</ol>
<p><strong>Equivalent condition:</strong> $\alpha(u + \lambda v) = \alpha(u) + \lambda \alpha(v)$ for all $u, v \in V$ and $\lambda \in F$.</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('48:02', 'Linear Map Examples', 'example', this)">
<span class="play-icon">▶</span>48:02
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<h3>Linear Map Examples</h3>
<p><strong>Matrix multiplication:</strong> For $A \in M_{m,n}(F)$, the map $\alpha: F^n \to F^m$ defined by $\alpha(v) = Av$ is linear.</p>
<p><strong>Differentiation:</strong> The map $D: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ defined by $D(f) = f'$ is linear.</p>
<p><strong>Integration:</strong> The map $I: C([0,1]) \to \mathbb{R}$ defined by $I(f) = \int_0^1 f(x) \, dx$ is linear.</p>
<p><strong>Evaluation:</strong> For $x_0 \in X$, the map $\text{ev}_{x_0}: F^X \to F$ defined by $\text{ev}_{x_0}(f) = f(x_0)$ is linear.</p>
<p><strong>Identity and zero maps:</strong> $\text{id}_V: V \to V$ and $0: V \to W$ are always linear.</p>
<p><strong>Composition:</strong> If $\alpha: V \to W$ and $\beta: U \to V$ are linear, then $\alpha \circ \beta: U \to W$ is linear.</p>
</div>
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('52:36', 'Non-Linear Examples', 'example', this)">
<span class="play-icon">▶</span>52:36
</div>
<h3>Non-Linear Examples</h3>
<p><strong>Squaring:</strong> The map $\alpha: \mathbb{R} \to \mathbb{R}$ defined by $\alpha(x) = x^2$ is not linear.</p>
<p><strong>Adding constant:</strong> The map $\beta: \mathbb{R} \to \mathbb{R}$ defined by $\beta(x) = x + 1$ is not linear.</p>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('53:06', 'End of Lecture Question', 'default', this)">
<span class="play-icon">▶</span>53:06
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<h3>Question for Next Time</h3>
<p>Consider the complex vector space $\mathbb{C}$ and the map $\alpha: \mathbb{C} \to \mathbb{C}$ defined by complex conjugation $\alpha(z) = \overline{z}$.</p>
<p><strong>Question:</strong> Is $\alpha$ linear?</p>
<p><em>Think about this over the weekend - we'll discuss it next time!</em></p>
</div>
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