Rank and Nullity

Every matrix $A$ hides important geometric information in its columns and solutions.

Rank

The Rank of a matrix is the number of linearly independent columns (or rows). It tells us the “true” dimension of the output space.

Null Space & Nullity

The Null Space is the set of all vectors $x$ such that $Ax = 0$ . The Nullity is the dimension of this Null Space.

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The Rank-Nullity Theorem

For an $m \times n$ matrix $A$ :

Rank(A) + Nullity(A) = n

This fundamental theorem creates a balance: The information preserved (Rank) plus the information lost (Nullity) equals the original input dimension ( $n$ ).