Glossary

3.2 Subspaces and their dimensions

4 min read•august 16, 2024

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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  • forms a subset of a maintaining vector space properties under addition and scalar multiplication
  • Contains the as a fundamental requirement
  • Demonstrates under vector addition and scalar multiplication
  • includes only the zero vector (0,0,0)
  • encompasses the entire vector space (R3\mathbb{R}^3)
  • Inherits parent vector space properties
    • Associativity: a+(b+c)=(a+b)+ca + (b + c) = (a + b) + c
    • Commutativity: a+b=b+aa + b = b + a
    • Distributivity: k(a+b)=ka+kbk(a + b) = ka + kb

Geometric Interpretation and Set Operations

  • Represents geometrically as lines, planes, or hyperplanes through origin
    • Line through origin in R2\mathbb{R}^2: y=mxy = mx
    • Plane through origin in R3\mathbb{R}^3: ax+by+cz=0ax + by + cz = 0
  • of subspaces always yields a subspace
    • Intersection of two planes in R3\mathbb{R}^3 results in a line subspace
  • of subspaces not guaranteed to be a subspace
    • Union of x-axis and y-axis in R2\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 uu and vv in subset SS, u+vu + v must also be in SS
  • Closure under scalar multiplication established by multiplying any subset vector by any scalar
    • For vector vv in subset SS and scalar cc, cvcv must be in SS
  • Counterexamples disprove subspace status by violating any of the three properties
    • Subset {(x,y)∣x>0}\{(x,y) | x > 0\} in R2\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}\{(x,y,z) | x + 2y - z = 0\} defines a plane subspace in R3\mathbb{R}^3
  • Inequalities typically do not define subspaces
    • {(x,y)∣x2+y2≤1}\{(x,y) | x^2 + y^2 \leq 1\} fails closure under scalar multiplication
  • of linear equations always define subspaces
    • Solutions to Ax=0Ax = 0 form the , a subspace of the domain
  • generally do not define subspaces
    • Solutions to Ax=bAx = b (where b≠0b \neq 0) fail to include zero vector

Dimension of a Subspace

Basis and Dimension Calculation

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

Dimension Relationships and Theorems

  • connects subspace dimensions in linear transformations
    • For T:V→WT: V \to W, dim(V)=dim(Im(T))+dim(Ker(T))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∩W)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 R2\mathbb{R}^2 yields entire R2\mathbb{R}^2 plane
  • occurs when subspace intersection is zero vector
    • R3=[span](https://www.fiveableKeyTerm:Span){(1,0,0)}⊕span{(0,1,0),(0,0,1)}\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 WW of VV, V/WV/W represents cosets of WW in VV

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
    • and are key subspaces in understanding transformations
  • Vector space decomposition into simpler subspace components
    • Eigenspace decomposition breaks space into invariant subspaces

Key Terms to Review (26)

Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space, meaning any vector in that space can be expressed as a linear combination of the basis vectors. The concept of basis is essential for understanding the structure and dimensionality of vector spaces, as well as the transformations that can be applied to them.
Closure: Closure refers to the property of a set, indicating that performing a certain operation on elements within that set will produce an element that also belongs to the same set. This concept is crucial in understanding both vector spaces and subspaces, as it ensures that operations like addition and scalar multiplication keep us within the boundaries of those sets. Without closure, the structure of vector spaces and their subspaces would not be well-defined, leading to inconsistencies in vector operations.
Column Space: The column space of a matrix is the set of all possible linear combinations of its column vectors. This space is essential for understanding the behavior of linear transformations and helps reveal important characteristics such as rank, which indicates the maximum number of linearly independent columns. The column space plays a critical role in determining the solutions to linear equations and is directly linked to the dimensions of subspaces.
Dimension: Dimension refers to the number of independent directions in a space, which is fundamentally connected to the concept of vector spaces. It tells us how many vectors are needed to form a basis for that space, indicating how many coordinates are required to represent points within it. This idea is crucial in understanding subspaces, as they can have dimensions that are less than or equal to the dimension of the entire space, and influences the properties of vector spaces themselves, including their representation using scalars, vectors, and matrices.
Direct Sum: The direct sum is a way to combine two or more subspaces into a new subspace, where every element can be uniquely expressed as the sum of elements from each subspace. This concept highlights how these subspaces interact, emphasizing their independence and the idea that their dimensions can be added together to give the dimension of the resulting space, provided that the intersection of the subspaces is only the zero vector.
Gaussian elimination: Gaussian elimination is a method used to solve systems of linear equations by transforming the augmented matrix into row-echelon form using a series of row operations. This technique helps to find solutions efficiently and reveals important properties of the matrix, such as rank and nullity, which are essential in understanding the structure of vector spaces and linear transformations.
Homogeneous Systems: Homogeneous systems refer to a set of linear equations in which all constant terms are equal to zero. These systems can be represented in matrix form as Ax = 0, where A is a matrix of coefficients and x is a vector of variables. A key feature of homogeneous systems is that they always have at least one solution, which is the trivial solution where all variables are zero.
Image: In linear algebra, the image of a linear transformation refers to the set of all output vectors that can be produced by applying the transformation to every vector in the input space. This concept is essential when understanding how transformations map vectors from one space to another, highlighting the nature of the output as a subspace of the target space. The image also plays a crucial role in determining properties like injectivity and the rank of the transformation.
Improper Subspace: An improper subspace is a specific type of subspace in linear algebra that includes the entire vector space itself. While proper subspaces contain fewer vectors than the full space, an improper subspace encompasses all possible vectors within the space, thereby not restricting its dimensions. Understanding this concept is crucial when discussing the dimensions and characteristics of vector spaces, as it highlights the boundaries of what can be considered a valid subspace.
Intersection: In the context of vector spaces, the intersection refers to the set of all vectors that belong to two or more subspaces simultaneously. This concept is crucial for understanding how different subspaces relate to each other, particularly in terms of their dimensions and the common elements they share. The intersection helps us identify shared properties between subspaces, leading to insights about their structure and dimensionality.
Kernel: The kernel of a linear transformation is the set of all vectors that are mapped to the zero vector by that transformation. It essentially captures the idea of how much information is lost when transforming data, providing insight into the relationship between the input and output spaces. The kernel is also a subspace, which means it can be analyzed in terms of dimension and properties similar to other subspaces in linear algebra.
Linear Independence: Linear independence refers to a set of vectors that do not express any vector in the set as a linear combination of the others. This concept is crucial because it determines whether a group of vectors can span a vector space or if they are simply redundant. Understanding linear independence helps in analyzing the structure of vector spaces, subspaces, and their dimensions, as well as establishing relationships between orthogonality, rank, and nullity.
Linear Transformation: A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you have a linear transformation, it will take a vector and either stretch, rotate, or reflect it in a way that keeps the relationships between vectors intact. Understanding how these transformations work is crucial in many areas like eigendecomposition, matrix representation, and solving problems in data science.
Non-Homogeneous Systems: Non-homogeneous systems are sets of linear equations that do not all equal zero, meaning at least one equation has a constant term on the right-hand side. This implies that the solutions to these systems represent a shift from the origin in a vector space, creating distinct characteristics in terms of their subspaces and dimensions. Understanding non-homogeneous systems is crucial because they can lead to unique solutions, no solutions, or infinitely many solutions depending on the relationships between the equations.
Null Space: The null space of a matrix is the set of all vectors that, when multiplied by that matrix, result in the zero vector. This concept is crucial in understanding solutions to linear equations, as it provides insight into the structure of a matrix and its transformations. The null space is closely linked to the rank of a matrix, as it helps determine the dimensions of subspaces associated with the matrix.
Nullity: Nullity is the dimension of the null space of a linear transformation or matrix, representing the number of linearly independent solutions to the homogeneous equation associated with that transformation. It measures the extent to which a linear transformation fails to be injective, revealing important insights about the relationships among vectors in vector spaces and their mappings.
Projection: Projection is a mathematical operation that transforms a vector into another vector that lies within a specified subspace, essentially representing the original vector in a simpler form. This concept plays a crucial role in various areas such as identifying orthogonal components and decomposing vectors, which is essential for understanding transformations and dimensional relationships in linear algebra.
Quotient Space: A quotient space is a construction in linear algebra and topology that partitions a vector space into disjoint subsets, called equivalence classes, based on a given equivalence relation. This concept allows us to simplify complex vector spaces by identifying certain vectors as equivalent, thus creating a new vector space whose structure reflects the relationships among the original vectors. Quotient spaces are particularly useful for understanding how subspaces relate to larger spaces, leading to insights about dimensions and linear transformations.
Rank: In linear algebra, rank is the dimension of the column space of a matrix, which represents the maximum number of linearly independent column vectors in that matrix. It provides insight into the solution space of linear systems, helps understand transformations, and plays a crucial role in determining properties like consistency and dimensionality of vector spaces.
Rank-Nullity Theorem: The Rank-Nullity Theorem states that for any linear transformation represented by a matrix, the sum of the rank and the nullity of the transformation equals the number of columns of the matrix. This theorem connects key concepts such as linear transformations, matrix representation, and subspaces, providing insight into how the dimensions of various vector spaces are related to each other.
Span: In linear algebra, the span of a set of vectors is the collection of all possible linear combinations of those vectors. This concept helps us understand how vectors can combine to fill out a space, making it crucial for grasping vector spaces, subspaces, and solving equations. By looking at the span, we can determine dimensions, identify dependencies between vectors, and understand their roles in creating solutions to linear systems.
Subspace: A subspace is a set of vectors that forms a vector space within a larger vector space, satisfying the same axioms and properties as the original space. It must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. Understanding subspaces helps in grasping important concepts like orthogonality, basis, dimension, and the structure of vector spaces.
Trivial subspace: A trivial subspace is the simplest type of subspace in a vector space, consisting solely of the zero vector. This concept connects to the broader idea of subspaces and their dimensions by highlighting that even the most basic vector space includes this fundamental element. Trivial subspaces serve as a starting point for understanding more complex subspaces, as every vector space must contain at least this one subspace to satisfy the properties that define vector spaces.
Union: In linear algebra, the union refers to the combination of two or more sets, where elements from each set are included without duplication. This concept is vital when discussing subspaces, as it helps in understanding how different subspaces can be combined and their collective structure in a vector space. The union allows for a more comprehensive view of how subspaces interact and overlap within a larger framework.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors that can be added together and multiplied by scalars, adhering to specific rules. It is fundamental in understanding linear combinations, linear independence, and spans, which are crucial for various applications in linear transformations, subspaces, and dimensional analysis.
Zero vector: The zero vector is a special vector that has all its components equal to zero, symbolically represented as $$ extbf{0}$$. It serves as the additive identity in vector spaces, meaning that when it is added to any other vector, the result is that same vector. The zero vector is essential for defining vector operations and for establishing the structure of both vector spaces and subspaces.
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