Algebra: Rank, Row, and Column Space

Algebra Notes

• The subject is algebra, taught by Associate Professor Nguyen Trung Thanh at the Faculty of Data Science and Artificial Intelligence.

Rank of a Matrix

• A matrix's rank is the dimension of its row space (or column space).
• It's denoted as rank(A).
• The rank of a matrix is equal to the dimension of its row space and its column space.

Column Space

• The column space (col(A)) of a matrix A is the set of all possible linear combinations of its columns.
• It's a subspace of ℝ^m.
• Example: If A has column vectors x₁ and x₂, then col(A) = Span{x₁, x₂}.

Row Space

• The row space (row(A)) of a matrix A is the set of all possible linear combinations of its rows.
• It's a subspace of ℝⁿ.
• Example: For a matrix A with rows y₁, y₂, y₃, the row space is row(A) = Span{y₁, y₂, y₃}.

Matrix Properties

• If matrix A transforms into B using elementary row operations, then row(A) = row(B).
• If matrix A transforms into B using elementary column operations, then col(A) = col(B).
• Nonzero rows of a row-echelon matrix R are a basis of row(R).
• Columns of R with leading ones are a basis of col(R).
• The row space and column space of an m × n matrix A have the same dimension (rank(A)).

Properties of Rank

• The rank of a matrix A is equal to the rank of its transpose A^T.
• The rank of a zero matrix is zero.
• The rank of an identity matrix of order n is n.
• The rank of matrix A is less than or equal to the minimum of m and n (where m is the number of rows and n is the number of columns).
• If U and V are invertible matrices, then rank(AU) = rank(VA) = rank(A).
• If A transforms into B by elementary row/column operations, then rank(A) = rank(B).
• The rank of the sum of two matrices A and B is less than or equal to the sum of their ranks.
• The rank of the product of two matrices AC is less than or equal to the minimum of rank(A) and rank(C).

Null Space

• The null space of a matrix A, denoted as null(A), is the set of all vectors x that satisfy Ax = 0.
• The dimension of the null space is called the nullity of A.

Image Space

• The image space of a matrix A, denoted by im(A), is the set of all possible outputs (Ax) when the matrix operates on all possible input vectors x.

Other Properties

• Rank(A) = n, if and only if the rows of A span ℝⁿ and the columns of A are linearly independent in ℝ^m.
• If the matrix A is an m × n matrix, and CA = I for an n × m matrix C, then the matrix A is invertible.
• If Ax = 0, and x is in Rⁿ, then x = 0.
• If m = n (square matrix), then det(A) ≠ 0 if and only if rank(A) = n.
• The system Ax = b is consistent for every b in ℝ^m.