There are two kinds of matrix spaces: one is defined by a set of matrices of special type (considered in section), and another is associated with every matrix. This section is devoted to the latter case.

Row spaces

Range or column space

Null space

Dimension Theorems

Four subspaces

Matrix Spaces

We know from previous discussions that every real or complex n-dimensional vector space is isomorphic to ℝn or ℂn. This gives us the ground to consider linear transformations from one Euclidean Vector Space ℝn (or ℂn) into another one ℝm (or ℂm) instead of general finite dimensional vector spaces. Every linear transformation T: V ↦ W from one n-dimensional vectors space V into another m-dimensional vector space W is equivalent to a corresponding matrix transformation x ↦ Ax for some m×n matrix A for appropriate chosen bases. The inverse statement is obviously true because multiplication by a matrix defines a linear transformation.

Let A be an m-by-n matrix that maps \( \mathbb{R}^n \) into \( \mathbb{R}^m . \) So for every (colomn) vector \( {\bf x} = \left[ x_1 , x_2 , \ldots , x_n \right]^{\mathrm{T}} \) corresponds a vector \( {\bf A}\,{\bf x} \in \mathbb{R}^m . \)

Let V and U be vector spaces and \( T:\,U \to V \) be linear transformation. Then the set of all vectors of the form \( {\bf y} = {\bf A}\,{\bf x} \in V \) is called the range (or image) of T.

Theorem: Let V and U be vector spaces and \( T:\,U \to V \) be linear transformation. Then the range of T is a subspace of V.

To clarify the notation, we use the symbols 0U and 0V to denote the zero vectors of U and V, respectively. We also denote by R(T) the range of linear transformation T.

Because \( T\left( 0U \right) = 0V, \) we have that \( 0V \in R(T) . \) Now let us choose two vectors \( {\bf x}, \, {\bf y} \in R(T) \) from the range and a scalar k. Then there exist two vectors u and v such that \( T({\bf u}) = {\bf x} \) and \( T({\bf v}) = {\bf y} . \) Hence, \( T({\bf u} + {\bf v}) = T({\bf u}) + T({\bf v}) = {\bf x} + {\bf y} \) and \( T(k{\bf u}) = k\, T({\bf u}) = k\, {\bf x} . \) Thus, \( T({\bf u} + {\bf v}) = T({\bf u}) + T({\bf v}) = {\bf x} + {\bf y} \in R(T) \) and \( T(k\,{\bf u}) = k \, T({\bf x}) = k\,{\bf x} \in R(T) , \) so R(T) is a subspace of V.

  Recall that a set of vectors β is said to generate or span a vector space V if every element from V can be represented as a linear combination of vectors from β.

Example: The span of the empty set \( \varnothing \) consists of a unique element 0. Therefore, \( \varnothing \) is linearly independent and it is a basis for the trivial vector space consisting of the unique element---zero. Its dimension is zero.

 

Example: In \( \mathbb{R}^n , \) the vectors \( e_1 [1,0,0,\ldots , 0] , \quad e_2 =[0,1,0,\ldots , 0], \quad \ldots , e_n =[0,0,\ldots , 0,1] \) form a basis for n-dimensional real space, and it is called the standard basis. Its dimension is n.

 

Example: Let us consider the set of all real \( m \times n \) matrices, and let \( {\bf M}_{i,j} \) denote the matrix whose only nonzero entry is a 1 in the i-th row and j-th column. Then the set \( {\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n \) is a basis for the set of all such real matrices. Its dimension is mn.

 

Example 4: The set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n \right\} \) form a basis in the set of all polynomials of degree up to n. It has dimension n+1. ■

 

Example 5: The infinite set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n , \ldots \right\} \) form a basis in the set of all polynomials. ■

 

A function T : D ↦ R, with domain set D and range set R, is said to be onto (surjective) if T(D) = R, that is, if \( R = \left\{ T(x) \,\big\vert \, x \in D \right\} . \)
A function T : D ↦ R, with domain set D and range set R, is said to be one-to-one (or injective) if it preserves distinctness: it never maps distinct elements of its domain to the same element of its range, that is, if whenever T(x) = T(y) for x,y ∈ D, then x = y.
A linear map T is a function from 𝔽n to 𝔽m that preserves linear combinations, and it is denoted as T : 𝔽n ⇾ 𝔽m. Thus, for any vectors x,y ∈ 𝔽n and any scalar 𝑎, b, we have T(𝑎x + b y) = 𝑎 T(x) + b T(y).
A function T : D ↦ R, with domain set D and range set R, is said to be bijection if T is both one-to-one and onto. A linear bijection T : V ↦ U from vector space V into another vector space U is called isomorphism.

Theorem 2: A linear map T : 𝔽n ⇾ 𝔽m is onto if Im(T) = 𝔽m, so dim(Im(T)) = m. This means that the corresponding m×n matrix is of full row rank (= its rows are linearly independent).

The onto condition follows automatically from the definition.

Theorem 3: A linear map T : 𝔽n ⇾ 𝔽m is one-to-one if ker(T) = {0}, and it is onto if Range(T) = 𝔽m.

Another way of saying this is that T is one-to-one if dim(ker(T)) = 0, and onto if dim(Image(T)) = m.
If T(x) = T(y) for x,y ∈ ℝn, then T(x) - T(y) = 0. However, since T is linear, this gives T(x-y) = 0. We can therefore conclude that x-y ∈ ker(T). So T is one-to-one exactly when ker(T) = {0}. The onto condition is automatic from the definition of onto and the range of the map.

Theorem 4: A linear map T : 𝔽n ⇾ 𝔽m is bijection if and only if T sends a basis of the domain 𝔽n to a basis of the range 𝔽m.

Corollary 1: A linear map T : 𝔽n ⇾ 𝔽m is bijection if and only if dim(ker(T)) = 0 and dim(Im(T)) = m.

Corollary 2: A linear map T : 𝔽n ⇾ 𝔽m is bijection if and only if n = m and the matrix A representing T is invertible.

Corollary 3: A linear map T : 𝔽n ⇾ 𝔽n is onto if and only if for any spanning subset S of 𝔽n, we have that T(S) is a spanning subset of 𝔽n.

Theorem 5: A linear map T : 𝔽n ⇾ 𝔽m is onto if Im(T) = 𝔽m, so dim(Im(T)) = m. This means that the corresponding m×n matrix is of full row rank (= its rows are linearly independent).

The onto condition follows automatically from the definition.