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?
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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?
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The rank equals the number of pivot columns.- What is rank(A) relative to rank(A^T)?
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rank(A) = rank(A^T); rank is unchanged by transpose.- Is row rank equal to column rank?
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Yes, the number of linearly independent rows equals the number of linearly independent columns.- What does full rank mean?
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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?
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min(m, n).- What is the rank of the zero matrix?
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0, since there are no pivot positions.- How do rank and invertibility relate for an n by n matrix?
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A square matrix is invertible if and only if rank(A) = n.- What inequality relates rank(AB) to rank(A) and rank(B)?
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rank(AB) <= min(rank(A), rank(B)).- In row echelon form, what counts toward rank?
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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?
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No, row operations preserve the rank.- What is the null space of a matrix A?
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The set of all vectors x such that Ax = 0.- What is the nullity of a matrix?
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The dimension of the null space.- How do free variables in Ax = 0 relate to nullity?
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The number of free variables equals the nullity of A.- What is a basis for the null space?
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A set of linearly independent vectors spanning the entire null space, typically one vector per free variable.
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