Ch 1.4 The Matrix Equation Ax = b

Flashcards covering definitions, theorems, and computational rules related to the matrix equation Ax = b, as discussed in the lecture notes.

Matrix-Vector Product Ax

If A is an m x n matrix with columns a₁, …, an, and x is in R^n, then Ax is the linear combination of the columns of A using the corresponding entries in x as weights: x₁a₁ + x₂a₂ + … + xn a_n.

When Ax is defined

The product Ax is defined only if the number of columns of A equals the number of entries in x.

Matrix Equation

An equation of the form Ax = b, where A is a matrix, x is a vector of unknowns, and b is a vector, which is equivalent to a system of linear equations or a vector equation.

Equivalence of Solution Sets (Theorem 3)

For an m x n matrix A, the matrix equation Ax = b, the vector equation x₁a₁ + … + xn an = b, and the system of linear equations with augmented matrix [a₁ … a_n b] all have the same solution set.

Existence of Solutions for Ax = b

The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.

Span R^m

A set of vectors {v₁, …, vp} in R^m spans (or generates) R^m if every vector in R^m is a linear combination of v₁, …, vp.

Conditions for A's columns to span R^m (Theorem 4)

For an m x n matrix A, the following are logically equivalent: 1) For each b in R^m, Ax = b has a solution. 2) Each b in R^m is a linear combination of the columns of A. 3) The columns of A span R^m. 4) A has a pivot position in every row.

Row-Vector Rule for Computing Ax

If the product Ax is defined, then the i-th entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.

Identity Matrix (I or I_n)

A square matrix with 1's on the main diagonal and 0's elsewhere. It acts as a multiplicative identity, meaning Ix = x for every vector x in R^n.

Properties of Matrix-Vector Product (Theorem 5)

For an m x n matrix A, vectors u and v in R^n, and a scalar c: a) A(u + v) = Au + Av; b) A(cu) = c(Au).