3.2 Subspaces and their dimensions

Vector spaces are like big playgrounds, and subspaces are special areas within them. These areas follow the same rules as the big playground but have their own unique features. They're crucial for understanding how vectors behave in different situations.

Subspaces come in various shapes and sizes, from simple lines to complex planes. By studying their dimensions and properties, we can solve tricky math problems and make sense of complicated data structures. It's all about breaking things down into manageable pieces.

Subspaces and their properties

Definition and Fundamental Properties

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• Subspace forms a subset of a vector space maintaining vector space properties under addition and scalar multiplication
• Contains the zero vector as a fundamental requirement
• Demonstrates closure under vector addition and scalar multiplication
• Trivial subspace includes only the zero vector (0,0,0)
• Improper subspace encompasses the entire vector space ($\mathbb{R}^3$)
• Inherits parent vector space properties
  • Associativity: $a + (b + c) = (a + b) + c$
  • Commutativity: $a + b = b + a$
  • Distributivity: $k(a + b) = ka + kb$

Geometric Interpretation and Set Operations

• Represents geometrically as lines, planes, or hyperplanes through origin
  • Line through origin in $\mathbb{R}^2$: $y = mx$
  • Plane through origin in $\mathbb{R}^3$: $ax + by + cz = 0$
• Intersection of subspaces always yields a subspace
  • Intersection of two planes in $\mathbb{R}^3$ results in a line subspace
• Union of subspaces not guaranteed to be a subspace
  • Union of x-axis and y-axis in $\mathbb{R}^2$ violates closure under addition

Identifying Subspaces

Subspace Verification Process

• Prove subset satisfies three defining properties for subspace classification
• Zero vector test confirms inclusion of parent space's zero vector in subset
• Closure under addition verified by showing sum of any two subset vectors remains in subset
  • For vectors $u$ and $v$ in subset $S$, $u + v$ must also be in $S$
• Closure under scalar multiplication established by multiplying any subset vector by any scalar
  • For vector $v$ in subset $S$ and scalar $c$, $cv$ must be in $S$
• Counterexamples disprove subspace status by violating any of the three properties
  • Subset $\{(x,y) | x > 0\}$ in $\mathbb{R}^2$ fails zero vector test

Analysis of Subset Definitions

• Equations often define subspaces (planes, lines through origin)
  • $\{(x,y,z) | x + 2y - z = 0\}$ defines a plane subspace in $\mathbb{R}^3$
• Inequalities typically do not define subspaces
  • $\{(x,y) | x^2 + y^2 \leq 1\}$ fails closure under scalar multiplication
• Homogeneous systems of linear equations always define subspaces
  • Solutions to $Ax = 0$ form the null space, a subspace of the domain
• Non-homogeneous systems generally do not define subspaces
  • Solutions to $Ax = b$ (where $b \neq 0$) fail to include zero vector

Dimension of a Subspace

Basis and Dimension Calculation

• Dimension equals number of vectors in subspace basis
• Basis comprises linearly independent set spanning the subspace
• Find basis by reducing spanning set to linearly independent set
  • Use Gaussian elimination or other reduction methods
• Rank of associated matrix equals subspace dimension
  • For matrix $A$, $rank(A)$ = dimension of column space of $A$
• Nullity defines dimension of matrix null space
  • For matrix $A$, $nullity(A)$ = dimension of null space of $A$

Dimension Relationships and Theorems

• Rank-nullity theorem connects subspace dimensions in linear transformations
  • For linear transformation $T: V \to W$, $dim(V) = dim(Im(T)) + dim(Ker(T))$
• Dimension comparison provides geometric insights
  • 1D subspace in 3D space represents a line
  • 2D subspace in 3D space indicates a plane
• Dimension formula for sum of subspaces
  • $dim(U + W) = dim(U) + dim(W) - dim(U \cap W)$

Vector Spaces vs Subspaces

Subspace Properties within Vector Spaces

• Vector spaces contain infinite subspaces including itself and trivial subspace
• Sum of subspaces creates smallest subspace containing both original subspaces
  • Sum of x-axis and y-axis in $\mathbb{R}^2$ yields entire $\mathbb{R}^2$ plane
• Direct sum occurs when subspace intersection is zero vector
  • $\mathbb{R}^3 = \text{[span](https://www.fiveableKeyTerm:Span)}\{(1,0,0)\} \oplus \text{span}\{(0,1,0),(0,0,1)\}$
• Quotient spaces formed by translations of subspace in parent space
  • For subspace $W$ of $V$, quotient space $V/W$ represents cosets of $W$ in $V$

Applications and Importance

• Fundamental subspaces of matrices have crucial relationships
  • Column space and row space dimensions are equal (rank of matrix)
  • Null space and left null space complement column and row spaces respectively
• Subspace analysis essential for solving linear equation systems
  • Homogeneous system solutions form null space subspace
• Linear transformations analyzed through subspace relationships
  • Kernel and image are key subspaces in understanding transformations
• Vector space decomposition into simpler subspace components
  • Eigenspace decomposition breaks space into invariant subspaces