Rank-nullity theorem: rank(A) + nullity(A) = n
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Invertible matrix
Rank-nullity theorem: rank(A) + nullity(A) = n
The rank-nullity theorem is a fundamental result in linear algebra that relates the dimensions of the row space, column space, and null space of a matrix.
The rank of a matrix, denoted as rank(A), is the dimension of its row space or column space. The nullity of a matrix, denoted as nullity(A), is the dimension of its null space. The theorem states that the sum of these two dimensions equals the number of columns n in the matrix.
Understanding this theorem helps in analyzing the structure of linear transformations represented by matrices. It provides insights into the solvability of linear systems and the properties of linear maps.
Example
Consider a 3x3 matrix A with rank(A) = 2. According to the rank-nullity theorem, nullity(A) must be 1 because rank(A) + nullity(A) = 3.
The rank-nullity theorem is crucial for determining the solvability of linear equations and understanding the behavior of linear transformations.
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