2.2 Bases and Dimension

Bases and dimension are key concepts in linear algebra. They help us understand vector spaces by providing a way to represent and measure them. A basis is a set of vectors that spans the space and is linearly independent.

The dimension of a vector space is the number of vectors in any basis. This concept connects to linear independence, as a basis must be linearly independent. It also relates to spanning, as a basis must span the entire space.

Basis of a Vector Space

Definition and Properties

• A basis for a vector space $V$ is a linearly independent subset of $V$ that spans $V$
  • Linear independence: No vector in the basis can be expressed as a linear combination of the other basis vectors
  • Spanning: Every vector in $V$ can be uniquely expressed as a linear combination of the basis vectors
• The coefficients in the linear combination are called the coordinates of the vector with respect to the basis
• A vector space can have multiple bases, but the number of vectors in any basis is always the same (this number is the dimension of the vector space)

Unique Representation and Coordinates

• Every vector in $V$ has a unique representation as a linear combination of basis vectors
  • For a basis $\{v_1, v_2, \ldots, v_n\}$ and a vector $u \in V$, there exist unique scalars $c_1, c_2, \ldots, c_n$ such that $u = c_1v_1 + c_2v_2 + \ldots + c_nv_n$
• The coefficients $c_1, c_2, \ldots, c_n$ are the coordinates of $u$ with respect to the basis $\{v_1, v_2, \ldots, v_n\}$
  • Coordinates provide a way to represent vectors in terms of the basis vectors
  • Different bases lead to different coordinate representations of the same vector

Identifying a Basis

Linear Independence

• A set of vectors $\{v_1, v_2, \ldots, v_n\}$ is linearly independent if the equation $c_1v_1 + c_2v_2 + \ldots + c_nv_n = 0$ has only the trivial solution $c_1 = c_2 = \ldots = c_n = 0$
  • Equivalently, no vector in the set can be expressed as a linear combination of the others
• To check linear independence, solve the equation $c_1v_1 + c_2v_2 + \ldots + c_nv_n = 0$ and verify that the only solution is the trivial solution
  • Example: For vectors $v_1 = (1, 0)$ and $v_2 = (0, 1)$, solve $c_1(1, 0) + c_2(0, 1) = (0, 0)$. The only solution is $c_1 = c_2 = 0$, so $v_1$ and $v_2$ are linearly independent

Spanning Property

• A set of vectors $\{v_1, v_2, \ldots, v_n\}$ spans a vector space $V$ if every vector in $V$ can be expressed as a linear combination of the vectors in the set
• To check if a set of vectors spans $V$, determine if every vector in $V$ can be written as a linear combination of the set
  • This can be done by solving a system of linear equations or using matrix methods (e.g., row reduction)
  • Example: To check if $\{(1, 0), (0, 1)\}$ spans $\mathbb{R}^2$, consider an arbitrary vector $(a, b) \in \mathbb{R}^2$ and solve $c_1(1, 0) + c_2(0, 1) = (a, b)$. The solution is $c_1 = a$ and $c_2 = b$, so the set spans $\mathbb{R}^2$

Basis Test

• A set of vectors $\{v_1, v_2, \ldots, v_n\}$ forms a basis for a vector space $V$ if and only if the vectors are linearly independent and span $V$
  • Both conditions must be satisfied for the set to be a basis
• To determine if a set is a basis, check linear independence and spanning property
  • Example: $\{(1, 0), (0, 1)\}$ is a basis for $\mathbb{R}^2$ because the vectors are linearly independent and span $\mathbb{R}^2$

Dimension and Basis Size

Definition of Dimension

• The dimension of a vector space $V$ is the number of vectors in any basis for $V$
  • All bases for a given vector space have the same number of vectors
• If a vector space has a basis with $n$ vectors, then the vector space is said to be $n$-dimensional
  • Example: $\mathbb{R}^3$ is a 3-dimensional vector space because the standard basis $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ has 3 vectors

Invariance of Dimension

• The dimension of a vector space is a well-defined concept and does not depend on the choice of basis
  • Any two bases for the same vector space have the same number of vectors
• Dimension is an intrinsic property of the vector space itself, not of any particular basis
  • Example: The standard basis $\{(1, 0), (0, 1)\}$ and the basis $\{(1, 1), (1, -1)\}$ for $\mathbb{R}^2$ both have 2 vectors, so $\mathbb{R}^2$ is 2-dimensional regardless of the basis chosen

Constructing a Basis

Extending a Linearly Independent Set

• To construct a basis for a vector space $V$, start with any linearly independent set of vectors in $V$ and extend it to a basis
  • Add vectors to the set until it spans the entire vector space while maintaining linear independence
• One method is to start with the empty set and add vectors one by one, checking linear independence at each step
  • Example: To construct a basis for $\mathbb{R}^3$, start with the empty set and add $(1, 0, 0)$, then $(0, 1, 0)$, and finally $(0, 0, 1)$. The resulting set $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ is a basis for $\mathbb{R}^3$

Gram-Schmidt Process

• The Gram-Schmidt process is a method for constructing an orthonormal basis from a linearly independent set of vectors
  • Orthonormal basis: A basis in which all vectors are orthogonal (perpendicular) to each other and have unit length
• The process involves iteratively subtracting the projection of each vector onto the previous orthogonal vectors and normalizing the result
  • Example: Applying the Gram-Schmidt process to the linearly independent set $\{(1, 0), (1, 1)\}$ in $\mathbb{R}^2$ yields the orthonormal basis $\{(1, 0), (0, 1)\}$

Row Reduction Method

• Another method for constructing a basis is to row reduce a matrix whose columns are the vectors in $V$
  • The columns corresponding to the pivot entries (leading 1s) in the reduced row echelon form of the matrix form a basis for $V$
• This method relies on the fact that the pivot columns are linearly independent and span the column space of the matrix
  • Example: To find a basis for the column space of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \end{pmatrix}$, row reduce the matrix to $\begin{pmatrix} 1 & 0 & -5 \\ 0 & 1 & 4 \end{pmatrix}$. The first two columns form a basis for the column space

Dimension Calculation from Basis

Counting Basis Vectors

• To calculate the dimension of a vector space $V$, find a basis for $V$ and count the number of vectors in the basis
  • The number of vectors in any basis is equal to the dimension of the vector space
• If a set of vectors $\{v_1, v_2, \ldots, v_n\}$ is known to be a basis for $V$, then the dimension of $V$ is $n$
  • Example: The set $\{1, x, x^2\}$ is a basis for the vector space of polynomials of degree at most 2, so the dimension of this space is 3

Rank-Dimension Theorem

• The dimension of a vector space can also be calculated by finding the rank of a matrix whose columns span the vector space
  • Rank of a matrix: The maximum number of linearly independent columns (or rows) in the matrix
• The rank-dimension theorem states that the rank of a matrix is equal to the dimension of its column space
  • Example: The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \end{pmatrix}$ has rank 2 (as seen from its reduced row echelon form), so the dimension of its column space is 2

Dimension of Specific Vector Spaces

• For certain vector spaces, the dimension can be determined without explicitly finding a basis
  • The dimension of the vector space of polynomials of degree at most $n$ is $n+1$
    • Example: The vector space of polynomials of degree at most 3 has dimension 4
  • The dimension of the vector space of $m \times n$ matrices is $mn$
    • Example: The vector space of $2 \times 3$ matrices has dimension 6
• Knowledge of the structure and properties of specific vector spaces can help determine their dimensions