# NotebookLM - Quotient Spaces ## Metadata **Status**:: #x **Zettel**:: #zettel/fleeting **Created**:: [[2025-11-19]] ## Synopsis The concept of Quotient Spaces, introduced in Chapter 3 of the sources along with Products of Vector Spaces and Duality, is a crucial and somewhat abstract topic in linear algebra. Here is a detailed breakdown to help you better understand Quotient Spaces: ### 1. Fundamental Definition via Translates The construction of a quotient space $V/U$ begins with the notion of a **translate**. 1. **Translate ($v+U$)**: If $v$ is a vector in $V$ and $U$ is a subset of $V$, the translate $v + U$ is defined as the subset of $V$ containing all possible sums of $v$ with elements from $U$: $v + U = \{v + u : u \in U\}$ (3.95, 82). 2. **Quotient Space ($V/U$)**: If $\mathbf{U}$ is a **subspace** of $V$, the quotient space $V/U$ is defined as **the set of all translates of $U$** (3.99, 83). $V/U = \{v + U : v \in V\}$. The elements of $V/U$ are not vectors in $V$, but rather *sets* (subsets of $V$), which leads to the abstraction often noted. **Key Property of Translates:** Two translates of a subspace $U$ are either equal or disjoint (3.101, 84). Crucially, $v + U = w + U$ if and only if $v - w \in U$ (3.101, 84). **Geometric Intuition (3.100, 83):** * If $U$ is a line in $\mathbf{R}^3$ containing the origin, $\mathbf{R}^3/U$ is the set of all lines in $\mathbf{R}^3$ that are parallel to $U$. * If $U$ is a plane in $\mathbf{R}^3$ containing the origin, $\mathbf{R}^3/U$ is the set of all planes in $\mathbf{R}^3$ parallel to $U$. * If $U$ is a line through the origin in $\mathbf{R}^2$, say $U = \{(x, 2x) \in \mathbf{R}^2 : x \in \mathbf{R}\}$, then a translate like $(17, 20) + U$ is the line in $\mathbf{R}^2$ that contains the point $(17, 20)$ and has slope 2. $V/U$ is the set of all lines in $\mathbf{R}^2$ with slope 2 (3.100, 83). ### 2. $V/U$ as a Vector Space For $V/U$ to qualify as a vector space (3.103, 85), addition and scalar multiplication must be defined on the set of translates: 1. **Addition**: $(v_1 + U) + (v_2 + U) = (v_1 + v_2) + U$. 2. **Scalar Multiplication**: $\lambda(v + U) = (\lambda v) + U$. The essential step in proving that $V/U$ is a vector space is verifying that these operations are **well-defined**. This means the result of the operation cannot depend on which representative vector ($v_1$ or $v_2$, etc.) is chosen for the translate. This relies directly on the property that $v_1 + U = v_2 + U \iff v_1 - v_2 \in U$. * The **additive identity** of $V/U$ is $0 + U$, which is simply the subspace $U$ itself. * The **additive inverse** of $v + U$ is $(-v) + U$. ### 3. Key Concepts and Results #### The Quotient Map ($\pi$) The **quotient map** $\pi: V \to V/U$ is a linear map defined by sending every vector $v$ in $V$ to its corresponding translate in $V/U$: $\pi(v) = v + U$ (3.104, 88). Using this map, two essential properties are established: * The **null space** of $\pi$ is exactly the subspace $U$ (3.105, 88). * The **range** of $\pi$ is the entire quotient space $V/U$ (3.105, 88). #### Dimension of the Quotient Space If $V$ is finite-dimensional and $U$ is a subspace of $V$, the dimension of the quotient space is given by a simple subtraction, relating the dimension of $V/U$ to the "lost" dimension of $U$: $\mathbf{\dim V/U = \dim V - \dim U}$ (3.105, 88). This result is proven directly by applying the Fundamental Theorem of Linear Maps to the quotient map $\pi$ (3.105). #### Isomorphism to the Range of a Linear Map One of the key uses of the quotient space construction is demonstrating a canonical isomorphism. For any linear map $T \in \mathcal{L}(V, W)$, the quotient space $V/(\text{null } T)$ is isomorphic to the image space $\text{range } T$ (3.107(d), 89). This shows that the quotient space $V/(\text{null } T)$ serves as a way to "mod out" the non-injective part of $T$. #### The Quotient Operator ($T/U$) When studying operators (linear maps from a vector space to itself), if $U$ is a subspace of $V$ that is **invariant under $T$** (meaning $T$ maps $U$ into $U$), we can define the **quotient operator** $T/U \in \mathcal{L}(V/U)$: $(T/U)(v + U) = Tv + U$. The condition that $U$ is invariant under $T$ is necessary to ensure that this definition is well-defined. The quotient operator $T/U$ is useful because it inherits properties from $T$; for instance, every eigenvalue of $T/U$ is also an eigenvalue of $T$. Also, if $T$ has an upper-triangular matrix representation, $T/U$ also has an upper-triangular matrix representation with respect to some basis of $V/U$. *** To solidify your understanding, think of a quotient space as a way to collapse an entire subspace $U$ down to a single point (the origin $0+U$) while studying the structure of $V$ relative to $U$. Every point in the quotient space $V/U$ represents a complete affine subspace (a translated copy of $U$) in the original space $V$.