Definitions

A matrix is said to be in row echelon form if

(i) The first nonzero entry in each nonzero row is 1. (ii) If row k does not consist entirely of zeros, the number of leading zero entries in row k + 1 is greater than the number of leading zero entries in row k. (iii) If there are rows whose entries are all zero, they are below the rows having nonzero entries.

reduced row echelon form

(i) The matrix is in row echelon form. (ii) The first nonzero entry in each row is the only nonzero entry in its column.

Determinants Row Operation I : Two rows of A are interchanged.

-det(A) Interchanging two rows (or columns) of a matrix changes the sign of the determinant.

Determinants Row Operation III : A multiple of one row is added to another row.

Adding a multiple of one row (or column) to another does not change the value of the determinant.

1. A + B = B + A 2. (A + B) + C = A + (B + C) 3. (AB)C = A(BC) 4. A(B + C) = AB + AC 5. (A + B)C = AC + BC 6. (αβ)A = α(βA) 7. α(AB) = (αA)B = A(αB) 8. (α + β)A = αA + βA 9. α(A + B) = αA + αB

Algebraic Rules

1. (AT )T = A 2. (αA)T = αAT 3. (A + B)T = AT + BT 4. (AB)T = BTAT

Algebraic Rules for Transposes

VECTOR AXIOMS: A1. x + y = y + x for any x and y in V. A2. (x + y) + z = x + (y + z) for any x, y, and z in V. A3. There exists an element 0 in V such that x + 0 = x for each x ∈ V. A4. For each x ∈ V, there exists an element −x in V such that x + (−x) =0. A5. α(x + y) = αx + αy for each scalar α and any x and y in V. A6. (α + β)x = αx + βx for any scalars α and β and any x ∈ V. A7. (αβ)x = α(βx) for any scalars α and β and any x ∈ V. A8. 1x = x for all x ∈ V.

C1. If x ∈ V and α is a scalar, then αx ∈ V. C2. If x, y ∈ V, then x + y ∈ V.

With each n × n matrix A it is possible to associate a scalar, det(A), whose value will tell us whether the matrix is nonsingular. Before proceeding to the general definition, let us consider the following cases.

Determinant

There are three types of elementary matrices corresponding to the three types of elementary row operations. Type I: An elementary matrix of type I is a matrix obtained by interchanging two rows of I. Type II: An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant. Type III: An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.

Elementary Matrices

The process of using row operations I, II, and III to transform a linear system into one whose augmented matrix is in row echelon form is called ____.

Gaussian elimination

Elementary Row Operations

I. Interchange two rows. II. Multiply a row by a nonzero real number. III. Replace a row by its sum with a multiple of another row.

A linear system is said to be ___ if there are more equations than unknowns. ____ systems are usually (but not always) inconsistent.

Overdetermined Systems

A system of m linear equations in n unknowns is said to be ____ if there are fewer equations than unknowns (m < n).

Underdetermined Systems

Two systems of equations involving the same variables are said to be ____ if they have the same solution set.

equivalent

A system of linear equations is said to be ____ if the constants on the right- hand side are all zero.

homogeneous

If a1, a2, ... , an are vectors in R^m and c1, c2, ... , cn are scalars, then a sum of the form c1a1 + c2a2 +···+ cnan is said to be a ____ of the vectors a1, a2, ... , an.

linear combination

An n × n matrix is said to be ____ if it does not have a multiplicative inverse.

singular

A system is said to be in ____ if, in the kth equation, the coefficients of the first k − 1 variables are all zero and the coefficient of xk is nonzero (k = 1, ... , n).

strict triangular form

If S is a nonempty subset of a vector space V, and S satisfies the conditions (i) αx ∈ S whenever x ∈ S for any scalar α (ii) x + y ∈ S whenever x ∈ S and y ∈ S then S is said to be a ____of V.

subspace

An n × n matrix A is said to be ____ if AT = A.

symmetric

Determinants Row Operation II : A row of A is multiplied by a nonzero scalar.

α det(A) Multiplying a single row or column of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.