Question Bank • QB_EXP_EM_SYS_002 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra: Systems of Linear Equations

Subtopic: Vector spaces and basis

Question Statement:

A homogeneous system of linear equations $Ax = 0$ with 4 variables has a coefficient matrix $A$ of rank 2. The dimension of the solution space (nullity of $A$) is equal to ________ (answer in integer).

Answer Key:

2

Step-by-Step Solution & Explanation

According to the Rank-Nullity Theorem for a linear transformation or system of equations with $n$ variables: $\operatorname{rank}(A) + \operatorname{nullity}(A) = n$ Given $n = 4$ and $\operatorname{rank}(A) = 2$: $\operatorname{nullity}(A) = n - \operatorname{rank}(A) = 4 - 2 = 2$ The dimension of the solution space is therefore $2$.
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