Rank, null space, and the rank-nullity theorem from a college linear algebra course.
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- What is the rank of a matrix?
- The dimension of its row space or column space, equivalently the number of pivot positions in row echelon form.
- How is rank related to pivot columns?
- The rank equals the number of pivot columns.
- What is rank(A) relative to rank(A^T)?
- rank(A) = rank(A^T); rank is unchanged by transpose.
- Is row rank equal to column rank?
- Yes, the number of linearly independent rows equals the number of linearly independent columns.
- What does full rank mean?
- The rank equals min(m, n), the smaller of the number of rows and columns.
- What is the maximum rank of an m by n matrix?
- min(m, n).
- What is the rank of the zero matrix?
- 0, since there are no pivot positions.
- How do rank and invertibility relate for an n by n matrix?
- A square matrix is invertible if and only if rank(A) = n.
- What inequality relates rank(AB) to rank(A) and rank(B)?
- rank(AB) <= min(rank(A), rank(B)).
- In row echelon form, what counts toward rank?
- Each row with a pivot, that is, a leading non-zero entry in a different column than the row above it.
- Does row reduction change the rank of a matrix?
- No, row operations preserve the rank.
- What is the null space of a matrix A?
- The set of all vectors x such that Ax = 0.
- What is the nullity of a matrix?
- The dimension of the null space.
- How do free variables in Ax = 0 relate to nullity?
- The number of free variables equals the nullity of A.
- What is a basis for the null space?
- A set of linearly independent vectors spanning the entire null space, typically one vector per free variable.
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