Summary

Matrix Space

Definition

Definition

Properties

• Matrix Space is a vector space over a field of matrices.
• The set of all $m\times n$ matrices with entries in a field $F$ is a vector space over $F$.
• The zero matrix is the additive identity.
• The additive inverse of a matrix is the matrix with all its entries negated.
• Scalar multiplication is defined by multiplying each entry of the matrix by the scalar.
• Matrix addition is defined by adding corresponding entries of the matrices.

Basis

Definition

A basis of a vector space $V$ is a set of linearly independent vectors that span $V$.

Dimension

Definition

The dimension of a vector space is the number of vectors in any basis for the space. The dimension of the vector space $V$ is denoted by $\dim (V)$.

Null Space

Definition

The null space of a matrix $A$ is the vector space construct using set of all vectors $x$ such that $Ax=0$ . It is denoted by $\text{null}(A)$ or $\text{ker}(A)$.

Definition

The nullity of a matrix $A$ is the dimension of the null space of $A$. It is denoted by $\text{nullity}(A)$ or $\text{dim}(\text{ker}(A))$.

Column Space

Definition

The column space of a matrix $A$ is the vector space construct using set of all linear combinations of the columns of $A$ $\left(\text{span}(c_1,c_2,\dots ,c_n)\right)$. It is denoted by $C_A$.

Remark

$C_A$ is a subspace of ${\mathbb{R}}^m$.

Remark

$C_A=\{y:\forall x,Ax=y\}$. This means that it is subspace of the result for the system of linear equations $Ax=y$.

Row Space

Definition

The row space of a matrix $A$ is the vector space construct using set of all linear combinations of the rows of $A$ $\left(\text{span}(r_1,r_2,\dots ,r_m)\right)$. It is denoted by $R_A$.

Remark

$R_A$ is a subspace of ${\mathbb{R}}^n$.

Theorem

$C_A=R_{A^T}$ and $R_A=C_{A^T}$ and $\text{dim}(C_A)=\text{dim}(R_A)$.

Rank

Definition

The rank of a matrix $A$ is the dimension of its column space. It is denoted by $\text{rank}(A)$.

Remark

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.

Theorem

$\text{rank}(A)=\text{rank}(A^T)$.

Rank-Nullity Theorem

Theorem

$\text{rank}(A)+\text{nullity}(A)=n$ and $\text{rank}(A^T)+\text{nullity}(A^T)=m$.