Finding Rank and Nullity of a Matrix

Given a 4x5 matrix $A = \begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$, find the rank and nullity of $A$.

When tasked with finding the rank and nullity of a matrix, key concepts from linear algebra come into play. The rank of a matrix is essentially the dimension of the column space, representing the maximum number of linearly independent column vectors in the matrix. A practical approach to identify this is to use row reduction to transform the given matrix into its row echelon or reduced row echelon form. The number of non-zero rows in this form will directly provide the rank of the matrix.

Nullity, on the other hand, pertains to the dimension of the kernel of the matrix. It counts the number of free variables when the matrix is viewed as a system of linear equations. Nullity is computed by subtracting the rank of the matrix from the total number of columns. Understanding how these concepts interconnect is crucial, as they tell us about the solutions to corresponding system of equations. Rank plus nullity meeting the total number of columns in the matrix serves as a pivotal theorem in this domain.

This problem of determining rank and nullity not only reinforces your skills in matrix reduction techniques but also provides a deeper appreciation of the relationships between different subspaces associated with a matrix, such as column space, null space, and how they manifest in dimensionality.