Rank of a Matrix

In the field of linear algebra, the rank of a matrix is a fundamental concept that helps us understand the properties and behavior of matrices. The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It gives us crucial information about the size of the largest non-zero minor in the matrix.

Definitions

1. Linearly Independent Vectors: Vectors are said to be linearly independent if no vector in the set can be represented as a linear combination of the others. In simpler terms, if you have a set of vectors and one of them can be expressed as a sum of the others multiplied by constants, then they are linearly dependent.
2. Non-zero Minor: A minor of a matrix is a determinant of a square submatrix obtained by deleting some rows and columns. A non-zero minor is a minor whose determinant is not zero.

Rank Theorem

For any matrix A, the rank of A is equal to the rank of its transpose.

Example

Consider the matrix: $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$

To find the rank of matrix A, we perform row operations to reduce it to row-echelon form: $\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Here, we see that there is only one linearly independent row. Thus, the rank of matrix A is 1.

Historical Context

The concept of rank was first introduced by the German mathematician David Hilbert in the late 19th century. It played a crucial role in the development of linear algebra and its applications in various fields such as physics, engineering, and computer science.

Real-life Example

In a system of linear equations, the rank of the coefficient matrix can tell us whether the system has a unique solution, infinitely many solutions, or no solution at all. This information is essential in various real-life scenarios, such as in image processing, control systems, and optimization problems.

Exam Questions

1. Calculate the rank of the matrix $B=\left[\begin{array}{ccc}2& 4& 6\\ 1& 2& 3\\ 3& 6& 9\end{array}\right]$
2. Discuss the significance of the rank of a matrix in the context of solving a system of linear equations.
3. Prove that the rank of a matrix is equal to the number of linearly independent columns in the matrix.