How to find rank and nullity of a matrix?
Understand the Problem
The question is asking for the processes or methods to determine the rank and nullity of a given matrix. It involves understanding the concepts of linear independence and the relationship between the dimensions of the column space and the null space of the matrix.
Answer
The rank is the number of leading 1's in RREF, and the nullity is calculated as $n - r$.
Answer for screen readers
The rank and nullity of the matrix can be computed as follows:
Let the rank be $r$ and the number of columns be $n$. The nullity is given by $n - r$.
Steps to Solve
- Find the rank of the matrix
To find the rank, you need to first convert the matrix into its reduced row echelon form (RREF) using row operations. The rank is the number of non-zero rows in this form.
- Convert the matrix to RREF
Apply elementary row operations (swap, multiply by non-zero scalar, and add/subtract rows) to make the leading coefficient (pivot) of each row equal to 1, and ensure that all entries below and above this leading 1 are zeros.
- Count the number of leading 1's
After obtaining the RREF, count how many leading 1's are present. This count gives you the rank of the matrix.
- Calculate the nullity of the matrix
The nullity can be computed using the formula: $$ \text{Nullity} = n - \text{Rank} $$ where $n$ is the total number of columns in the matrix.
- Summarize the relationship between rank and nullity
Notice that rank plus nullity will equal the number of columns in the matrix: $$ \text{Rank} + \text{Nullity} = n $$
This relationship is known as the Rank-Nullity Theorem.
The rank and nullity of the matrix can be computed as follows:
Let the rank be $r$ and the number of columns be $n$. The nullity is given by $n - r$.
More Information
The Rank-Nullity Theorem is a fundamental result in linear algebra that describes the dimensions of a linear map's kernel (null space) and image (column space). This concept is key to understanding the structure of linear transformations and their associated matrices.
Tips
- Confusing row rank with column rank. Remember that row rank and column rank are always equal.
- Not simplifying the matrix completely to RREF before counting. Ensure all operations follow the rules of RREF.
- Forgetting to account for all columns when calculating nullity. Always use the total number of columns in the matrix in your calculations.
