1. Row and column vector:

Let A be a matrix of order m×n. Then there are m rows and n columns. Each row gives a matrix of order 1×n and is known as a row vector. Similarly, each column gives a n×1 matrix and is known as a column vector. A row vector can be considered as an element of Rⁿ and column vector can be considered as an element of R^m.

2. Null space

2.1. Null space of a linear transformation:

Let V and W be vector spaces over the field R and let T:V→W be a linear transformation. Then the null space of T is defined as the set of vectors x∈V such that T(x)=0 and is also known as kernel of T. It is denoted by N(T) or ker(T). Thus N(T)={ x∈V|T(x)=0}.

2.2. Null space of a matrix:

Null space of a matrix A of order m×n consists of all the column vectors X such that AX=0.

2.3. Nullity:

Dimension of the null space of a linear transformation T (or a matrix A) is called the nullity, which is denoted by η(T) (or η(A)).

2.4. Examples:

(i). Consider the linear transformation T:R²→R² such that T(x, y)=(x, -y), where x, y∈R. Then nullity of T is zero because its null space is {(0, 0)}.
(ii). Consider the linear transformation T:R²→R² such that T(x, y)=(x, 0), where x, y∈R. Then nullity of T is one because its null space is {(0, α): α∈R}, where α∈R.
(iii). Nullity of the zero transformation is 0.
(iv). Let A= be a matrix of order 2×2. Then nullity of A is zero because its null space is {(0, 0
(v). Let A=, where a, b are non-zero real numbers, be a matrix of order 2×2. Then nullity of A is two because its null space is {(a, b): a, b∈R}.

3. Range:

Let V and W be vector spaces over a field R. Let T:V→W be a linear transformation. Then the range of T is defined as set of vectors α∈W such that α=T(x), for some x∈V which is denoted by R(T). Thus R(T)={T(x)∈V : x∈V}.

3.1. Rank:

Dimension of range of a linear transformation T is called the rank, which is denoted by ρ(T). Rank of a matrix A is the maximal number of linearly independent row or column vectors which is denoted by ρ(A).

3.2. Remark:

Rank (or nullity) of a matrix is same as the rank (or nullity) of the linear transformation associated with the matrix and vice-versa.

3.3. Examples:

(i). Consider the linear transformation T:R²→R² such that T(x, y)=(x, -y), where x, y∈R. Then rank of T is 2 because its range is {(α, β): α, β∈R}.
(ii). Consider the linear transformation T:R²→R² such that T(x, y)=(x, 0), where x, y∈R. Then rank of T is one because its range is {(α, 0): α∈R}.
(iii). Rank of the zero transformation is 0.
(iv). Let A=

$$\begin{pmatrix}a & b \\\ c & d\end{pmatrix}$$

be a matrix of order 2×2. Then rank of A is 0 because set of row vectors is {(0, 0)} which is linearly dependent.
(v). Let A=

$$\begin{pmatrix}0 & 0 \\\ 0 & 0\end{pmatrix}$$

where a, b are non-zero real numbers, be a matrix of order 2×2. Then nullity of A is two because its null space is {(a, b): a, b∈R}.

4. Rank-Nullity theorem:

Let V and W be finite dimensional vector spaces over R and let T:V→W be a linear transformation. Then ρ(T)+η(T)=Dim V.

4.1. Example:

Let T:R²→R³ be a map defined as T(x, y)=(x, y, x+y). We find the rank and nullity of T. One can easily verify that T is a linear transformation. Now, consider the standard basis of R². Then T(1, 0)=(1, 0, 1) and T(0, 1)=(0, 1, 1). Thus the range(T) is a subspace of R³ spanned by the vectors (1, 0, 1) and (0, 1, 1). These vectors are linearly independent which implies that rank(T)=2. By rank-nullity theorem, ρ(T) + η(T) =Dim(R²). This implies that η(T)=2-2=0. Thus nullity of T is zero. Alternatively, to find the nullity, notice that null space T= {(0, 0)}.