Rank-Nullity Theorem Proof

Statement: Let U and V be vector spaces over the field F and let T be a Linear Transformation (L.T.) from U into V. Suppose that U is finite-dimensional. Then, rank (T) + nullity (T) = dim U.

Proof:

Let N be a null space of T, then N is a subspace of U. Since U is finite-dimensional. Therefore, N is finite-dimensional.

Let dim N = nullity(T) = K and let {α₁, α₂,….., α_k} be a basis for N

∵ {α₁, α₂,….., α_k} is a linearly independent subset of U

∴ We can extend it to form a basis of U

Let dim U = n and let {α₁, α₂,….., α_k, α_k+1, α_k+2,……, α_n} be a basis of U

Vectors T(α₁),…., T(α_k), T(α_k+1), T(α_k+2),……., T(α_n) are in the range of T

We shall show that {T(α_k+1), T(α_k+2),…., T(α_n)} is a basis for the range of T

(I) First, we shall prove that the vectors T(α_k+1), T(α_k+2),….., T(α_n) span the range of T

Let β ∈ range of T, then ∃ α ∈ U such that T(α) = β

Now α ∈ U ⇒ ∃ a₁, a₂,……, a_n ∈ F such that

α = a₁α₁ + a₂α₂+……+ a_nα_n

⇒ T(α) = T(a₁α₁+a₂α₂+……+a_nα_n)

⇒ T(α) = T(a₁α₁+a₂α₂+……+a_kα_k+a_k+1α_k+1+……+a_nα_n)

⇒ β = a₁ T(α₁)+……+a_k T(α_k)+ a_k+1 T(α_k+1)+……+ a_n T(α_n)

⇒ β = a_k+1 T(α_k+1) + a_k+2 T(α_k+2) +…..+ a_n T(α_n)

[∵ α₁, α₂,….., α_k ∈ N ⇒ T(α₁) = 0,…., T(α_k) = 0]

∴ the vectors T(α_k+1), T(α_k+2),……, T(α_n) span the range of T.

(II) Now we shall show that the vectors T(α_k+1), T(α_k+2),……, T(α_n) are L.I.

Let c_k+1, c_k+2,…, c_n ∈ F such that

c_k+1T(α_k+1) + c_k+2T(α_k+2) +…..+ c_nT(α_n) = 0

⇒ T(c_k+1α_k+1 + c_k+2α_k+2 +……+ c_nα_n) = 0

⇒ c_k+1α_k+1, c_k+2α_k+2,….., c_nα_n ∈ null space of T, i.e., N

⇒ c_k+1α_k+1 + c_k+2α_k+2 +…….+ c_nα_n = b₁α₁ + b₂α₂ +…+ b_kα_k

for some b₁,b₂,….., b_k ∈ F.

[∵ Each vector in N can be expressed as a linear combination of the vectors α₁,……, α_n forming a basis of N]

⇒ b₁α₁ + b₂α₂ +…..+ b_kα_k – c_k+1α_k+1 – c_k+2α_k+2 -…….- c_nα_n = 0

⇒ b₁ = b₂ = …… = b_k = c_k+1 = c_k+2 = ……. = c_n = 0

[∵ α₁, α₂,…., α_k, α_k+1,……, α_n are linearly independent being basis of U]

⇒ Vectors T(α_k+1), T(α_k+2),….., T(α_n) are linearly independent.

∴ vectors T(α_k+1), T(α_k+2), …., T(α_n) form a basis of the range of a T.

∴ Rank T = Dim range of T = n – k

Hence proved

Rank and Nullity are essential concepts in linear algebra, particularly in the context of matrices and linear transformations. They help describe the number of linearly independent vectors and the dimension of the kernel of a linear mapping.

In this article, we will learn what Rank and Nullity, the Rank-Nullity Theorem, and their applications, advantages, and limitations.