1. (Replacement) Replace one row by the sum of itself and a multiple of another row

2.  (Interchange) Interchange two rows

3.  (Scaling) Multiply all entries by a nonzero constant

1.  All nonzero rows are above any rows of all zeros.

2.  Each leading entry of a row is in a column to the right of the leading entry of the row above it. 

3.  All entries in a column below a leading entry are zero.

1.  The leading entry in each nonzero row is 1.

2.  The leading 1 is the only nonzero entry in its column.

Each matrix is row equivalent to one and only reduced echelon form

Existence and Uniqueness Theorem

A linear system is consistent iff the rightmost column of the augmented matrix is not a pivot column (iff an echelon form of the augment matrix has no row of the form [0 0 0..b] with b nonzero)

Parallelogram Rule for Addition

If u + v in R^2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u, o, and v

Algebraic Properties of R^n

For all u, v, w in R^n and all scalars c and d

i. u + v = v + u

ii.  (u +v) + w = u + (v + w)

iii.  u + 0 = 0 + u = u

iv.  u + (-u) = -u + u = 0, where -u denotes (-1)u

v.  c(u + v) = cu + cv

vi.  (c + d)u = cu + du

vii.  c(du) = (cd)u

viii.  1u = u

A vector equatin x1a1 + x2a2 + ... + xnan = b has the same solution set as the linear system whose augmented matrix is [a1 a2 ... an b]

b can be generated by a linear combo of a1...an iff there exists a solution to the linear system corresponding to [a1 a2 ...an b]

If v1,...,vp are in R^n, then the set of all linear combos of v1,...,vp is denoted by Span {v1,...,vp} and is called the subset of R^n spanned by v1,...,vp

Span {v1,...,vp} is the collection of all vectors that can be written in the form c1v1 + c2v2 + ... + cpvp with c1...cp scalars

If A is a mxn matrix w/ columns a1,...,an and if x is in R^n then the product of A and x, denoted by Ax, is the linear combo of the columns of A denoted w/ the corresponding entries in x as weights

ie. Ax = [a1 a2 ... an][x1...xn] = x1a1 + x2a2 + ... + xnan

(vertical column)^

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

Theorem 4 (1.4)

If A is an mxn matrix, with columns a1...an, and if b is in R^m, the matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ... + xnan = b which also has the same solution set of the system of linear equations whose augmented matrix is [a1 a2 ... an b]

Let A be a mxm co-efficient matrix.  All the following statements are all true or all false.

1.  For each b in R^m, the equation Ax = b has a solution.

2.  Each b in R^m ia linear combo of the columns of A.

3.  The column of A span R^m.

4.  A has a pivot position in every row.

Identity Matrix

a matrix with 1's on the diagonal and 0's elsewhere

Theorem 5 (1.4)

If A is a mxn matrix, u and v are vectors in R^n, and c is a scalar, then:

a.  A(u + v) = Au + Av

b.  A(cu) = c(Au)

A system of linear equations is said to be homogeneous if...

it can be written in the system Ax = 0, where A is a mxn matrix and 0 is the zero vector in R^m.

Ax = 0 always has at least one solution -> x = 0.

The zero vector is called the trivial solution

The solution set of a homogenous equation Ax = 0 can always be expressed as Span {v1,...,vp} for the suitable vectors v1,...,vp (v1,...,vp are the vectors from the parametric vector form of the solution)

If the only solution is the zero vector, then the solution set is Span {0}

If the equation Ax = 0 has only one free variable, the solution set is a line through the origin

Parametric Vector Equation

x = su + tv (s,t in R and u and v as vectors)

Theorem 6 (1.5)

Suppose the equation Ax = b is consistent for some given b, and let p be a solution.  Then the solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0.

Only applicable if Ax = b has at least one nonzero solution p. If not, the solution set is empty

In indexed set of vectors {v1,...,vp} in R^n is said to be linearly independent if the vector equation x1v1 + x2v2 + ...xpvp = 0 has only the trivial solution

linearly independent = only the trivial solution

The set {v1,...,vp} is said to be linearly dependent if there exist weights c1,...,cp, not all zero, such that c1v1 + c2v2 + ...cpvp = 0

linearly dependent = nontrivial solution/free variable

c1v1 + c2v2 + ... + cpvp = 0 is called a linear dependence relation among v1,...,vp when the weights are not all zero

The columns of the matrix A are linearly independent iff the equation Ax = 0 has ONLY the trivial solution

A set of two vectors {v1, v2} is linearly dependent if at least one of the vectors is a multiple of the other. (In geometric terms, if the vectors lie on the same line through the origin)

The set is linearly independent iff neither of the vectors is a multiple of the other

Theorem 7: Characterization of Linearly Dependent Sets

An indexed set (A set) S = {v1,...,vp} of two or more vectors is linearly dependent iff at least one of the vectors in S is a linear combination of the others.

Every vector does not need to be a linear combo of the preceding vectors.

Theorem 8 (1.7)

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1,...,vp} in R^n is linearly dependent if p>n

ie.  If number of columns > number of rows

Theorem 9 (1.7)

If a set S = {v1,...,vp} in R^n contains the zero vector, then the set is linearly dependent

In terms of linear transformations:

Solving Ax = b is the same as finding all the vectors x in R^n that are transformed into the vector b in R^m under the "action" of multiplication by A

A transformation/function/mapping T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(x) in R^m.  The set R^n is called the domain of T, and R^m is called the codomain of T. 

T : R^n -> R^m indicates that the domain is R^n and the codomain is R^m

For all x in R^n, the vector T(x) in R^m is called the image of x (under the action of T)

The set of all images T(x) is called the range of T.

The matrix transformation x |-> Ax is for each x in R^n, T(x) = Ax, where A is a mxn matrix.

The domain of T is R^n when A has n entries and the codomain of T is R^m when A has m entries.

The range of T is the set of all linear combinations of the columns of A b/c each T(x) is of the form Ax.

A transformation (or mapping) of T is linear if:

1.  T(u + v) = Tu + Tv

2.  T(cu) = cT(u) for all u and all scalars c

Every matrix transformation is a linear transformation

If T is a linear transformation, then...

1. T(0) = 0

2.  T(cu + dv) = cT(u) + dT(v) for all vectors u,v in the domain of T and all scalars c,d.

Let T: R^n -> R^m be a linear transformation.  Then there exists a unique matrix A such that

T(x) = Ax for all x in R^n

In fact, A is the m x n matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix in R^n

A = [T(e1) T(e2) ... T(en)] : standard matrix for the linear transformation T

Theorem 11 (1.9)

Let T : R^n -> R^m be a linear transformation.  Then T is one-to-one iff the equation T(x) = 0 has only the trivial solution.

Theorem 12 (1.9)

Let T : R^n -> R^m be a linear transformation and let A be the standard matrix for T.  Then: 

a.  T maps R^n onto R^m iff the columns of A span R^m

b.  R is one-to-one iff the columns of A are linearly independent.

Theorem 1 (2.1)

a.  A + B = B + A -> only defined if A and B are the same size

b.  (A + B) + C = A + (B + C)

c.  A + 0 = A

d.  r(A + B) = rA + rB

e.  (r + s)A = rA + sA

f.  r(sA) = (rs)A 

All verified by showing that the matrix on the left side has the same size as the matrix on the right and that corresponding columns are equal.

If A is a mxn matrix, and B is an nxp matrix with columns b1...bp, then the product AB is the mxp matrix whose columns are Ab1,...,Abp.  That is,

AB = A[b1 b2 ... bp] = [Ab1 Ab2 ...Abp]

This makes A(Bx) = (AB)x for all x in R^p

Let A be a mxn matrix and let B adn C have sizes for which the indicated sums and products are defined.

a. A(BC) = (AB)C

b.  A(B + C) = AB + AC

c.  (B + C)A = BC + BA

d.  r(AB) = (rA)B = A(rB) for any scalar r

e.  ImA = A = AIn

In general,

1.  AB does not = BA

2.  If AB = AC, then B does not = C

3.  If AB = 0, cannot conclude A or B = 0.

Theorem 3 (2.1)

Let A and B denote matrices whose sizes are appropriate for the following sums and products.

a.  (A^T)^T = A

b.  (A + B)^T = A^T + B^T

c.  For any scalar r, (rA)^T = rA^T

d.  (AB)^T = B^T x A^T

An nxn matrix A is said to be invertible if there is an nxn matrix C such that CA = I and AC = I, where I = In, the nxn identity matrix, in which case C is an inverse of A.

A matrix that is invertible is sometimes called a nonsingular matrix, and a matrix that is not invertible is sometimes called a singular matrix

Theorem 4 (2.2)

Let A = [a/c b/d] where / indicates verticalness.  If ad - bc does not equal 0, then A is invertible and

A^(-1) = (1/ad-bc)[d/-c -b/a]

If ad - bc = 0, then A is not invertible.

Theorem 5 (2.2)

If A is an invertible nxn, then for each b in R^n, the equation Ax = b has the unique solution x =A^(-1)b 

Theorem 6 (2.2)

a.  If A is an invertible matrix, then A^(-1) is invertible and (A^(-1))^(-1) = A

b.  If A and B are nxn invertible matrices, then so is AB and the inverse of AB is the product of the inverses of A and B in the reverse order, or (AB)^(-1) = B^(-1) x A^(-1)

c.  If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^(-1), or (A^T)^(-1) = (A^(-1))^T

An elementary matrix is...

a matrix that is obtained by performing a single elementary row operation on the identity matrix

Theorem 7 (2.2)

An nxn matrix A is invertible iff A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In, also transforms In into A^(-1).

The Invertible Matrix Theorem

Let A be a square nxn matrix.  Then the following statements are all true or all false.

a.  A is an invertible matrix

b.  A is row equivalent to the nxn identity matrix

c.  A has n pivot positions

d.  The equation Ax = 0 has only the trivial solution

e.  The columns of A form a linearly independent set.

f.  The linear transformation x |-> Ax is one-to-one

g.  The equation Ax = b has at least one solution for each b in R^n.

h.  The columns of A span R^n.

i.  The linear transformation x |-> Ax maps R^n onto R^n

j.  There is an nxn matrix C such that CA = I.

k.  There is a nxn matrix D such that AD = I

L.  A ^T is an invertible matrix

m.  The columns of A form a basis of R^n

n.  Col A = R^n

o.  dim Col A = n

p. rank A = n

q.  Nul A = {0}

r. dim Nul A = o

s.  The number 0 is not an eigenvalue of A.

t.  The determinant of A is not zero.

Theorem 9 (2.3)

Let T : R^n -> R^n be a linear transformation and let A be the standard matrix for T.  Then T is invertible iff A is an invertible matrix.  In that case, the linear transformation S given by S(x) = A^(-1)x is the unique function that satisfies S(T(x)) = x and T(S(x)) = x for all x in R^n

A linear transformation T: R^n -> R^n is said to be invertible if there exists a function S : R^n -> R^n such that

S(T(x)) = x for all x in R^n

T(S(x)) = x for all x in R^n

If such an x exists, it is unique and a linear transformation.

S is the inverse of T and can be written as T^(-1)

A subset of R^n is any set H in R^n that has three properties:

a.  The zero vector is in H.

b.  For each u and v in H, the sum u + v is in H.

c. For each u in H, and each scalar c, the vector cu is in H.

The column space of a matrix A is the set Col A of all linear combinations of the columns of A 

The column space of a mxn matrix is a subspace of R^m

When a system of linear equations is written in the form Ax = B, the column space of A is the set of all b for which the system has a solution.

Theorem 12 (2.8)

The null space of an mxn matrix is a subspace of R^n. Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of R^n.

To test whether a given vector v is in Nul A, compute Av to see if Av is the zero vector.

To create an explicit description of Nul A, solve the equation Ax = 0 and write the solution in parametric vector form.

A basis for a subspace H is a linearly independent set in H that spans H.

The Rank Theorem

If a matrix A has n columns, then rank A + dim Nul A = n

The Basis Theorem

Let H be a p-dimensional subspace of R^n.  Any linearly independent set of exactly p elements in H us automatically a basis for H.  Also, any set of p elements of H that spans H is automatically a basis for H.

Theorem 2 (3.1)

If a is a triangular matrix, then det A is the product of the entries on the main diagonal of A. 

Theorem 3 (3.2)

Row Operations

Let A be a square matrix.

a.  If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A

b.  If two rows are interchanged to produce B, then det B = - det A

c.  If one row of A is multiplied by k to produce B, then det B = k x det A

det A = (-1)^r x the product of the pivots of U, where r is the number of row interchanges and U is the echelon form of A, when A is invertible

det A = 0 when A is not invertible

Theorem 4 (3.2)

A square matrix A is invertible iff det A is not equal to 0.  

Theorem 5 (3.2)

If A is an nxn matrix, then det A^T = det A

Theorem 6 (3.2)

If A and B are nxn matrices, then det (AB) = det A x det B

Note: det (A + B) does not generally equal det A + det B

If A is an nxn matrix and E is an nxn elementary matrix, then det EA = det E x det A

where

det E = 1 if E is a row replacement, -1 if E is an interchange, or r if E is a scale by r