Matrices Flashcards

Pivot Columns in Matrices

• A 4 × 5 matrix can have at most 4 pivot columns, limited by the number of rows.
• Pivot columns contain leading non-zero entries in each row of its echelon form.

Invertible Matrices

• If A and B are invertible n × n matrices, the product AB is also invertible; however, (AB) ^−1 = B ^−1 A ^−1, not A ^−1 B ^−1.

Consistency of Equations with Invertible Matrices

• For any invertible n × n matrix A, the equation Ax = b is consistent for every vector b in R^n.
• If Ax = ej is consistent for every j (1 ≤ j ≤ n), then A must be invertible.

Relations Between Matrices

• If AB = AC and A is invertible, then B must equal C.
• The equality A TA = AAT does not hold for all n × n matrices A.

Matrix Transposition

• For vectors u and v in R^n, the matrices uvT and vuT are transposes of each other.
• Transposition properties: (AB) T = B TA T and (A T) T = A.

Identity Matrix in Reduced Echelon Form

• An n × n matrix A with n pivot positions has its reduced echelon form as the identity matrix I_n.

LU Decomposition

• In the LU decomposition of an m × n matrix A, L is an m × m lower triangular matrix with ones on the diagonal; U is the echelon form of A with pivots on the diagonal.

Four Fundamental Subspaces

• The four subspaces related to matrices are:
  • Column Space (C(A))
  • Null Space (N(A))
  • Row Space (C(A^T))
  • Left Null Space (N(A^T))

Column Space

• The column space C(A) is the span of the pivot columns of matrix A and is a subspace of R^m.
• If Ax = b is solvable, then vector b belongs to C(A).

Null Space

• The null space N(A) is defined as the set of all vectors x such that Ax = 0; it forms a subspace of R^n.
• N(A) = {0} when there is a pivot in each column.

Row Space

• The row space of a matrix A, denoted C(A^T), is the span of the rows of A and is a subspace of R^n.

Left Null Space

• The left null space N(A^T) consists of all solutions x such that x^T A = 0, forming a subspace of R^m.

Matrix Transpose Properties

• Transposition flips the matrix across its main diagonal, defined as (A^T)_ij = A_ji.
• Key properties include: (A^T)^T = A, (A+B)^T = A^T + B^T, and (AB)^T = B^TA^T.

Inverse Matrices

• An n × n matrix A has an inverse if it has a pivot in every row and column, leading to a unique solution for Ax = 0.
• The inverse is computed using the relationship A^-1 A = I and the property (AB) ^-1 = B^-1 A^-1.