MATH 304 Texas A&M Dr. Pitts Midterm

inconsistent

a system of linear equations is inconsistent if it has no solution

consistent

a system of linear equations is consistent if it has at least 1 solution

solution set

the set of all solutions to a system of linear equations

equivalence

two systems of linear equations are equivalent if they involve the same variables and have the same solution set

strict triangular form

each equation has exactly 1 more zero coefficient before the first nonzero coefficient than the equation above it and the first equation has no zero coefficients

elementary row operations (3)

I. The orders of any two equations may be interchanged.

II. Both sides of an equation may be multiplied by the same nonzero real number

III. A multiple of one equation may be added to (or subtracted from) another

coefficient matrix

a matrix formed from a system of linear equations

lead variables

the first nonzero element in a row in reduced row echelon form

free variables

the variables corresponding to the columns skipped during row reduction (i.e. the columns without lead variables in them)

overdetermined system

a system with more equations than unknowns.

usually but not always inconsistent

underdetermined system

a system with fewer equations than unknowns.

usually but not always consistent with infinitely many solutions. cannot have a unique solution

row echelon form (3)

I. the first nonzero entry in each nonzero row is 1

II. if a row is nonzero, the number of leading zeroes in it is strictly fewer than the number of leading zero entries in the row after it.

III. if there are zero rows, they are below all nonzero rows.

Gaussian elimination

the use of row operations to turn a matrix into row echelon form

reduced row echelon form

row echelon form with the additional constraint that the first nonzero entry in each row must be the only nonzero entry in its column

Gauss-Jordan reduction

the use of row operations to turn a matrix into reduced row echelon form

homogeneous

a system of linear equations is homogeneous if the constants on the right hand side are all 0 (i.e. Ax = 0).

In an m x n homogeneous system, if there is a unique solution, it is the zero vector. If n > m, there will always be free variables, and thus additional nontrivial solutions.

row vector

1 x n matrix

column vector

n x 1 matrix, also just called a vector

Euclidean n-space

set of all n x 1 column vectors of real numbers ℝⁿ

linear combination

representation of Ax as x_1 a_1 + x_2 a_2, where a_n is the nth column of A and x_n the nth element of the column vector x

transpose

The transpose of the m x n A is the n x m B where b_ji = a_ij. Denoted by Aᵀ.

symmetric

A matrix A is symmetric if A = Aᵀ.

nonsingular/invertible

An n x n matrix A is invertible if there exists a matrix B such that AB = BA = I.

identity matrix

The n x n matrix where I_ij = 1 if i=j, else 0. Each column in I is denoted e_j.

singular

An n x n matrix A is singular if it is not invertible.

algebraic rules for transposes (4)

I. (Aᵀ)ᵀ = A

II. (rA)ᵀ = rAᵀ

III. (A + B)ᵀ = Aᵀ + Bᵀ

IV. (AB)ᵀ = BᵀAᵀ

row equivalence

Matrices A and B are row equivalent if A can be transformed to B through elementary row operations.

elementary matrix

a matrix created by performing exactly 1 row operation on I

Vector Space Axiom 1

x + y = y + x for any x and y in V

Vector Space Axiom 2

(x + y) + z = x + (y + z) for any x, y, and z in V

Vector Space Axiom 3

There exists an element 0 in V such that x + 0 = x for each x in V.

Vector Space Axiom 4

For each x in V, there exists an element -x in V such that x + (-x) = 0.

Vector Space Axiom 5

r(x + y) = rx + ry for each scalar r and any x and y in V.

Vector Space Axiom 6

(r + s)x = rx + sx for any scalars r and s and any x in V.

Vector Space Axiom 7

(rs)x = r(sx) for any scalars r and s and any x in V.

Vector Space Axiom 8

1x = x for all x in V.

closure properties (2)

If x is in V and r is a scalar, then rx is in V.

If x, y are in V, then x + y is in V.

vector space C[a, b]

The set of all real-valued functions that are defined and continuous on the closed interval [a, b].

vector space Pn

The set of all polynomials of degree less than n.

additional properties of vector spaces (3)

I. 0x = 0

II. Iff x + y = 0, y = -x

III. (-1)x = -x

subspace

If S is a nonempty subset of V, and S satisfies the conditions:

I. rx is in S for all x in S and any scalar r
II. x + y is in S for all x, y in S

then S is a subspace of V.

null space

The set of all solutions of the homogeneous system Ax = 0.

The null space is a subspace of Rn.

spanning set

a set of vectors in a vector space V such that a linear combination of the vectors can form any vector in V

linear independence

a set of vectors is linearly independent if no vector in the set can be written as a linear combination of any of the others

linear dependence

a set of vectors is linearly dependent if a vector in the set can be written as a linear combination of the others.

If x1, x2, ..., xn are vectors in Rn and X = (x1 ... xn), the vectors will be linearly dependent iff X is singular.

basis

The vectors v1...vn form a basis for V iff v1...vn are linearly independent and span V.

dimension

The dimension of a vector space V is the number of vectors in its basis.

finite dimensional

A vector space has a finite dimension

infinite dimensional

A vector space does not have a finite dimension