Linear algebra theorems

LINEAR ALGEBRA THEOREMS

A homogeneous system of m equations in n variables either has a unique solution (zero solution) or has infinitely many solutions, depending on whether the number of leading 1s in the reduced row-echelon form is equal to n or is less than n (unique where 1s = n)

Assuming that the sizes of the matrices are such that the indicated operations can be performed the following rules hold: -

A + B = B + A - Commutativity of matrix addition

A + (B+C) = (A+B) + C - Associativity of matrix addition

AB ≠ BA

A(BC) = (AB)C

A(B+C) = AB + AC (A+B)C = AC + BC

a(B+C) = aB + aC

(a+b)C = aC + bC

(ab)C = abC

a(BC) = (aB)C = B(aC)

where A, B, C are matrices and a, b, c are scalars

Assuming the size of the matrices are such that the following operations can be performed: -

Add zero  A + 0 = 0 + A = A

Subtracting  A A = 0

Multiplying  A0 = 0, 0A = 0

Every system of linear equations has either no solution (lines are parallel), exactly one solution (lines all cross in same place), or infinitely many solutions (same line)

If A is invertible then A has a UNIQUE inverse, A^-1 and the determinant of A ≠ 0

If A, B are invertible matrices of the same size then AB is invertible with inverse B^-1A^-1 an extension of this is A₁.A_k are invertible and the same size then (A₁.A_k) is invertible with inverse (A^-1_k.A^-1₁)

If A is a square matrix and k, l are positive integers then

A^k.A^l = A^k+l

(A^k) ^l = A^kl

If A is invertible then: -

A^-1 is invertible and (A^-1) ^-1 = A

Aⁿ is invertible and (Aⁿ) ^-1 = (A^-1) ⁿ = A^-n

For any non-zero scalar k, then (kA) is invertible and (kA) ^-1 = 1/k A^-1

Multiplying i^th row of A by C is equivalent to multiplying matrix A on the left by the matrix obtained by multiplying the i^th row of I by c. Interchanging rows i and j of A is equivalent to multiplying A on the left by the matrix obtained by interchanging the i^th and j^th rows of I. Adding a multiple c of the i^th row of A to the j^th row of A is equivalent to multiplying A on the left by the matrix obtained by adding c i^th row of I to j^th row of I.

Every elementary matrix is invertible and the inverse is also an elementary matrix

If A is a square nn matrix then the following are equivalent: -

a) A is invertible

b) AX = 0 has only the zero solution

c) The reduced row echelon form of A is I_n

If A is an invertible nn matrix then for each n1 matrix B, the system of equations AX = B has exactly one solution, namely X = A^-1 B

Let A be a square matrix, if B is a square matrix satisfying BA = I then A is invertible and B=A^-1 (same applies if AB = I)

(recall theorem 1.11) The following are equivalent: -

a) A is invertible

d) AX=B is consistent for every n1 matrix B

If u, v, w are vectors in 2 space or 3 space and k and l are real scalars then the following rules apply: -

a)      u + v = v + u

b)      (u+v) + w = u + (v+w)

c)      u + 0 = 0 + u = u by definition

d)      u + (-u) = 0

e)      k(lu) = (kl)u

f)       k(u+v) = ku + kv

g)      (k+l)u = ku + lu

h)      1. u = u

If u and v are vectors in 2 or 3 space then u.v = ||u|| ||v|| cos θ

If u and v are vectors then v.v = ||v|| and the angle between u and v is acute iff u.v>0, obtuse iff u.v < 0 and π/2 iff u.v=0

If u and v are vectors and k is a real scalar then: -

a)      u v = v. u

b)      u (v+w) = u.v + u.w

c)      k(u.v) = (ku).v = u .(kv)

d)      v.v = 0 unless v = 0

If u, a are non-zero vectors then proj_a u) = (u.a)/ ||a|| . a

The basic rules of vector arithmetic in Rⁿ: -

a)      u + v = v + u

b)      (u+v) + w = u + (v+w)

c)      u + 0 = 0 + u = u

d)      u + (-u) = 0

e)      k(lu)= (kl)u

f)       k(u+v) = ku + kv

g)      (k+l)u = ku + lu

h)      1.u = u

Same as 2.4

Let V be a vector space and u is a vector in V and k a scalar then: -

a)      0.u = 0

b)      k0 = 0

c)      (-1)u = -u

d)      if ku=0 then k = 0 or u = 0

If W is a non-empty subset of V then W is a subspace iff: -

a)      u and v are in W, then u + v is also in W

b)      if k is a scalar, then ku is in W

If v₁,.,v_r are vectors in vector space V then the set of all linear combinations of v₁,.,v_r is a subspace of V, called W

A set S with two or more vectors is: -

a)      Linearly dependent iff one of the vectors in S is expressible as a linear combination of the others

b)      Linearly independent iff no vector in S is expressible as a linear combination of the other vectors in S

Let S = be a set of vectors in R^m. If n>m then S is linearly dependent

If S = is a basis for V then every set with more than n vectors is linearly dependent

Any two basis for a finite dimensional vector space V has the same number of vectors

a) If S = is linearly independent in an n-dimensional vector space V, then S is a basis for V

b) If S = is a spanning set for a n-dimensional vector space V, then S is a basis for V

c)      If S = is a linearly independent set in a n-dimensional vector space, V, r<n then S can be enlarged to a basis of V

Elementary row operations do not change the row space of a matrix

The non-zero row vectors in the (reduced) row echelon form of A form a basis for the row space of A

If A is any matrix, then the row space of A and the column space of A have the same dimension

(look at 1.14) If A is an nn matrix, the following are equivalent: -

a)      A is invertible

b)      AX=0 has only the trivial solution (i.e. the null space of A is )

c)      A is row equivalent to the identity

d)      AX = b is consistent for every n1 matrix b

e)      A has rank n

f)       The row vectors of A are linearly independent

g)      The column vectors of A are linearly independent

If S = is a basis for a vector space V, then every vector in V can be expressed uniquely as a linear combination of the vectors in S

If P is a transition matrix from a basis B to a basis B then P is invertible and P-1 is the transition matrix from B to B  [v]_B = P^-1 [v]_B

If T:V to W is a linear transformation: -

a)      T0 = 0

b)      T(-v) = -T(v)

If T:V to W is a linear transformation then: -

a)      ker(T) is a subspace of V

b)      im(T) is a subspace of W

If T:V to W is a linear transformation from an n-dimensional vector space to a vector space W then rank(T) + nullity(T) = n

If T:Rⁿ to R^m is a linear transformation and if is the standard basis for Rⁿ then T is given by multiplication by A, where A is the matrix whose successive columns are Te₁,.,Te_n

Let T:V to V be a linear transformation on a finite dimensional vector space V. If A is the matrix of T w.r.t some basis B and A is the matrix of T w.r.t a basis B then A = P^-1AP where P is the transition matrix from B to B

Let T₁:U to V and T₂:V to W be linear transformations. Let B, B, B be bases of U, V, W respectively. If A denotes the matrix for T₁ w.r.t B and B and C denotes the matrix for _T2 w.r.t Band B then CA is the matrix for T_{2 ˚} T₁ w.r.t B and B

Suppose D: M_nn(R) to R is alternating. Then if A is a matrix with two identical rows then D(A) = 0

Suppose D is a determinant function. Let B be a matrix obtained from A by performing an elementary row operation of adding a multiple of one row to another, then D(B) = D(A)

Let D: M_nn(R) to R be a determinant fuction, then if: -

a)      B is obtained from A by multiplying a row by k then D(B) = kD(A)

b)      B is obtained from A by interchanging two rows then D(B) = -D(A)

c)      B is obtained from A by adding a multiple of one row to another then D(B) = D(A)

Let D: M_nn(R) to R be a determinant function. If A = (a_ij) is a upper of lower triangular matrix then D(A) = a₁₁.a₂₂.a_nn (diagonal entries)

There is a unique determinant function, if it exists

An nn matrix A is invertible iff the determinant ≠ 0

If E is an elementary matrix then det(EA) = det(E)det(A)

Let A,B be any two nn matrices then D(AB)=D(A)D(B)

For any nn matrix A, D(A) = D(A^t)

Let D: M_nn(R) to R be defined by det A) = ∑ sign(σ) a_1σ(1)a_nσ(n). It is n linear, alternating and det(I) = 1

The determinant can be computed by the formula:

Det A) = a_1jC_1j + a_2jC_2j++a_njC_nj = co-factor expansion along column j

Det A) = a_i1C_i1 + a_i2C_i2++a_inC_in = co-factor expansion along row i

If A is invertible then A^-1 is given by:

A^-1 = 1/det(A) . adj A)

Cramers Rule If AX=B is a system of linear equations where A is an invertible nn matrix then the unique solution to the system is given by X = 1/det(A) . detA₁

detA_n

where A_j is a matrix obtained from A by replacing jth column by B

If A is an nn matrix the following are equivalent: -

a)      λ is an eigenvalue of A

b)      The homogeneous system of equations (A- λI)x = 0 has a non-zero solution

c)      There is a non-zero vector x in Rⁿ s.t Ax = λx

d)      λ is a real solution to the characteristic equation det(A- λI) = 0

A matrix A is diagonalized iff A has n linearly independent eigenvectors

If v₁,.,v_k are eigenvectors of A corresponding to distinct eigenvalues, λ₁,., λ_k then is a linearly independent set

If A is an nn matrix with n distinct eigenvalues then A is diagonalizable

The characteristic equation of A is (-1)^N(λ^N (c₁λ^N+1 + + c_N)) = 0