Study Notes on Linear Dependence and Independence

Linear Independence and Linear Dependence

The set of vectors S = {x₁, x₂, …, x_n} in a vector space V is said to be linearly independent provided that the equation

c₁x₁ + c₂x₂ + … + c_nx_n = 0

has only the trivial solution c₁ = c₂ = … = c_n = 0

Example-1: The standard unit vectors in ℜⁿ, viz.,

are linearly independent.

Note:

Any subset of a linearly independent set is a linearly independent set.

The coefficients in a linear combination of the vectors in a linearly independent set are unique.

A set of vectors is called linearly dependent if it is not linearly independent.

Example-2: The vectors u = (1, -1, 0), v = (1, 3, -1), and w = (5, 3, -2) are linearly dependent since 3u + 2v - w = 0.

Exercise: Determine whether the following vectors in ℜ⁴ are linearly dependent or independent.

(1, 3, -1, 4), (3, 8, -5, 7), (2, 9, 4, 23).

The vectors x₁, x₂, …, x_n are linearly dependent if and only if at least one of them is a linear combination of the others.

Theorem: The n vectors x₁, x₂, …, x_n in ℜⁿ are linearly independent if and only if the n x n matrix

A=[x₁ x₂ … x_n]

with the vectors as columns has nonzero determinant.

Theorem: Consider k vectors in ℜⁿ, with k < n. Let

A=[x₁, x₂, …, x_k]

be the n × k matrix having the k vectors as columns. Then the vectors x₁, x₂, …, x_k are linearly independent if and only if some k × k submatrix of A has nonzero determinant.

A finite set S of vectors in a vector space V is called a basis for V provided that

• The vectors in S are linearly independent;
• The vectors in S span V.

Example-3: The set of standard unit vectors in ℜⁿ, viz.,

form the standard basis for ℜⁿ.

Note: Any set of n linearly independent vectors in ℜⁿ is a basis for ℜⁿ.

Example-4: u = (-2, 1, 0, 0) and v = (1, 0, 1, 1) is a basis of the solution space of the homogeneous system given in Example 3. The dimension of the solution space is 2.

Theorem: Let S = {x₁, x₂, …, x_n} be a basis for the vector space V. Then any set of more than n vectors in V is linearly dependent.

Theorem: Any two bases of a vector space consist of the same number of vectors.

A vector space V is called finite dimensional if it has a basis consisting of a finite number of vectors. The unique number of vectors in each basis for V is called the dimension of V and is denoted by dim(V).

A vector space that is not finite dimensional is called infinite-dimensional.

Let  be a matrix. The row vectors of A are the m vectors in ℜⁿ given by

The subspace of ℜⁿ spanned by the m row vectors r₁, r₂, .., r_m is called the row space of the matrix A and is denoted by Row(A).

The dimension of the row space Row(A) is called the row rank of the matrix A.

Theorem: The nonzero row vectors of an echelon matrix are linearly independent and therefore form a basis for its row space Row(A).

Theorem: If two matrices are equivalent, then they have the same row space.

Note: To find a basis for the row space of a matrix, reduce the matrix to echelon form. Then the nonzero row vectors of the echelon matrix form a basis for the row space.

Let  be a matrix. The column vectors of A are the n vectors in ℜ^m given by

The subspace of ℜ^m spanned by the n column vectors c₁, c₂, …, c_n is called the column space of the matrix A and is denoted by Col(A).

The dimension of the row space Col(A) is called the column rank of the matrix A.

Note: To find a basis for the column space of a given matrix, reduce the matrix to echelon form. Then the column vectors of the given matrix that correspond to the pivot columns of the echelon matrix form a basis for the column space.

Exercise: Find a basis for the row space of the following matrix and state the dimension (row rank) of the row space. Also find a basis for the column space of the given matrix and state the dimension (column rank) of the column space.

Theorem: The row rank and the column rank of any matrix are equal.

The common value of the row rank and column rank of a matrix is called the rank of the matrix.

The solution space of the homogeneous system Ax = 0 is called the null space of A, denoted by Null(A)

Note:

1. If A is an m x n matrix, then Null(A) and Row(A) are subspaces of ℜⁿ, whereas
   Col(A) is a subspace of ℜ^m.
2. rank(A) + dim Null(A) = n.

Application: Consider a homogeneous system of m linear equations in n unknowns (m ≤ n). If the m × n coefficient matrix has rank r, (so r out of the m equations are irredundant), the system has n - r linearly independent solutions.

Theorem: The nonhomogeneous linear system Ax = b is consistent if and only if the vector b is in the column space of A.