### Problem StatementIf the rank of a $ 4 \times 7 $ matrix $ A $ is 2, what is the dimension of the solution space $ Ax = 0 $?**The dimension of the solution space is** $\_\_\_\_\_\_ $.---### ExplanationIn linear algebra, the dimension of the solution space of a homogeneous system of linear equations, $ Ax = 0 $, can be determined using the rank-nullity theorem. The rank-nullity theorem states:\[ \text{rank}(A) + \text{nullity}(A) = n \]where $ n $ is the number of columns in the matrix $ A $, the rank of $ A $ is the dimension of the column space (or the number of linearly independent columns), and the nullity of $ A $ is the dimension of the null space (or the solution space of $ Ax = 0 $).Given:- The rank of matrix $ A $ is 2.- The matrix $ A $ has 7 columns.We need to find the dimension of the solution space, which is the nullity of $ A $.Using the rank-nullity theorem:\[ \text{rank}(A) + \text{nullity}(A) = n \]\[ 2 + \text{nullity}(A) = 7 \]Solving for nullity:\[ \text{nullity}(A) = 7 - 2 = 5 \]Thus, **the dimension of the solution space is 5**.

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If the rank of a 4x7 matrix A is 2, what is the dimension of the solution space Ax = 0? The dimension of the solution space is

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Problem Statement

If the rank of a $ 4 \times 7 $ matrix $ A $ is 2, what is the dimension of the solution space $ Ax = 0 $?

**The dimension of the solution space is** $\_\_\_\_\_\_ $.

---

### Explanation

In linear algebra, the dimension of the solution space of a homogeneous system of linear equations, $ Ax = 0 $, can be determined using the rank-nullity theorem.

The rank-nullity theorem states:
\[ \text{rank}(A) + \text{nullity}(A) = n \]
where $ n $ is the number of columns in the matrix $ A $, the rank of $ A $ is the dimension of the column space (or the number of linearly independent columns), and the nullity of $ A $ is the dimension of the null space (or the solution space of $ Ax = 0 $).

Given:
- The rank of matrix $ A $ is 2.
- The matrix $ A $ has 7 columns.

We need to find the dimension of the solution space, which is the nullity of $ A $.

Using the rank-nullity theorem:
\[ \text{rank}(A) + \text{nullity}(A) = n \]
\[ 2 + \text{nullity}(A) = 7 \]

Solving for nullity:
\[ \text{nullity}(A) = 7 - 2 = 5 \]

Thus, **the dimension of the solution space is 5**.

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