lin alg midterm 1

Invertible Matrix Theorem

Let A be a square n × n matrix. Then the following are equivalent:

(i) A is an invertible matrix.

(ii) A is row equivalent to the n × n identity matrix.

(iii) A has n pivots positions.

(iv) The equation Ax = 0 has only the trivial solution.

(v) The columns of A form a linearly independent set.

(vi) The linear transformation x ↦ Ax is one-to-one.

(vii) The equation Ax = b has at least one solution for each b in Rn.

(viii) The columns of A span Rn.

(ix) The linear transformation x ↦ Ax maps Rn onto Rn.

(x) There is an n × n matrix C such that CA = I.

(xi) There is an n × n matrix D such that AD = I.

(xii) A^T is an invertible matrix.

Echelon form

A rectangular matrix is in echelon form (or row echelon form) if it has the following properties:

(i) All nonzero rows are above any rows of all zeros.

(ii) Each (non-zero) leading entry of a row is in a column to the right of the leading entry of the row above it.

(iii) All entries in a column below a leading entry are zeros.

Reduced Row Echelon Form

If a matrix in echelon form satisfies the following additional conditions, then it is in

reduced echelon form (or reduced row echelon form):

(iv) The leading entry in each nonzero row is 1.

(v) Each leading 1 is the only nonzero entry in its column.

Homogenous Linear System

A system of linear equations is said to be homogeneous if it can be written in the form

Ax = 0, where A is an m × n matrix and 0 is the zero vector in Rm.

Such a system Ax = 0 always has at least one solution, namely, x = 0 (this zero vector is in Rn), called the trivial solution.

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.