MAT223 Definitions

the pivot of a row…

is the leftmost non-zero entry in that row

a matrix is in row echelon form if…

1. all rows consisting only of zeroes are at the bottom

2. the pivot of each non-zero row in the matrix is in a column to the right of the pivot of the row above it

a matrix is in reduced row echelon form if…

1. the matrix is in echelon form

2. the pivot of each non-zero row is 1

3. each pivot is the only non-zero entry in its column

a system of linear equations is called consistent…

if it has at least one solution. if the system has no solutions, it is called inconsistent

let C be the coefficient matrix for a consistent system of linear equations in variables x₁, …, x_n.

we say that x_i is a basic variable for the system if…

we say that x_i is a free variable for the system if…

the i^th column of RREF(C) has a pivot

if the i^th column of RREF(C) does not have a pivot

Rouche-Capelli Theorem:

suppose a system of linear equations has augmented matrix A and coefficient matrix C. then…

1. the system is inconsistent if and only if the last column of RREF(A) has a pivot

2. the system has exactly one solution if and only if the last column of RREF(A) does not have a pivot, and every column of RREF(C) has a pivot

3. the system has infinitely many solutions if and only if the last column of RREF(A) does not have a pivot and RREF(C) has a column without a pivot

a linear combination of vectors v₁, v₂, …, v_n in R^m is…

a vector of the form w = c₁v₁ + c₂v₂ + … + c_nv_n where c₁, c₂, …, c_n are called the coefficients of the linear combination

the span of vectors v₁, v₂, …, v_n in R^m is…

the set Span(v₁, …, v_n) = {c₁v₁ + c₂v₂ + … + c_nv_n | c₁, c₂, …, c_n in R}. that is, Span(v₁, …, v_n) is the set of all linear combinations of vectors v₁, v₂, …, v_n

a set of vectors {v₁, v₂, …, v_n} in R^m is called linearly dependent if…

at least one of the vectors is a linear combination of the others. that is, for at least one i in {1, …, n}, we have V_i in Span(v₁, v₂, …, v_i-1, v_i+1, …, v_n). otherwise, the vectors are linearly independent

a vector space is any set of vectors V in R^m that satisfies all of the following properties:

1. V is non-empty (i.e. the zero-vector is in V)

2. V is closed under vector addition: for every v, w in V, v + w is in V.

3. V is closed under scalar multiplication: for every v in V and real number c, cv is in V

let V be a vector subspace of Rⁿ. a spanning/generating set for V is…

any subset B of V such that V=Span(B)

a subset B of a vector space V is called a basis if…

1. B is a spanning set for V

2. B is linearly independent

let V be a non-zero vector subspace of Rⁿ. then, the dimension of V, dim(V), is…

the size of any basis for V

the standard basis for Rⁿ is the set ε := {e₁, e₂, …, e_n} where e_i is…

the vector with 1 in the i^th coordinate and 0 in all other coordinates

let A be a mxn matrix with column vectors A = [v₁ v₂ … v_n]. then, for every vector x in Rⁿ, the matrix-vector product of A and x is…

the vector in R^m defined by Ax := x₁v₁ + … + x_nv_n

let A be a mxn matrix. then, the matrix transformation associated to A is the function:

T_A: Rⁿ→R^m defined by T_A(x) := Ax

a function F: Rⁿ→R^m is called linear if it satisfies the following properties for all vectors v, w in Rⁿ and scalars c in R:

1. F(v + w) = F(v) + F(w)

2. F(cv) = cF(v)

let F: Rⁿ→R^m be a linear transformation. then the defining matrix of F is the mxn matrix M satisfying:

F(x) = Mx, for all vectors x in Rⁿ, denoted M = M_F

a function f: X→Y is called injective if:

for every y in Y, there is at most one input x in X such that f(x) = y

a function f: X→Y is called surjective if:

for every y in Y, there is at least one input x in X such that f(x) = y

a function f: X→Y is called bijective if:

f is both injective and surjective

let V be a subspace of Rⁿ and W a subspace of R^m. an isomorphism between V and W is…

any linear bijective map F: V→W. if an isomorphism exists between two vector spaces, we say these spaces are isomorphic, and write V ≅ W

let F: Rⁿ→R^m. the kernel of F is the subset ker(F)⊆Rⁿ defined by:

ker(F) := {x in Rⁿ | F(x) = 0}

let F: Rⁿ→R^m. the image of F is the subset im(F)⊆R^m defined by:

im(F) := {F(x) | x in Rⁿ}

let F: Rⁿ→R^m. the rank of F is…

the dimension of im(F), denoted rank(F)

let F: Rⁿ→R^m. the nullity of F is…

the dimension of ker(F), denoted nullity(F)

let A be a mxn matrix with column vectors A = [v₁ … v_n]. then, the column space of A is…

the subspace of R^m given by Col(A) := Span(v₁, …, v_n)

let A be a mxn matrix with column vectors A = [v₁ … v_n]. then, the null space of A is…

the subspace of Rⁿ given by Nul(A) := {x in Rⁿ | Ax = 0}

let A be a matrix. the nullity of A is…

the dimension of Nul(A), denoted nullity(A)

let A be a matrix. the rank of A is…

the dimension of Col(A), denoted rank(A)

a system of linear equations is called homogeneous if:

the constant coefficients are all equal to zero

let A = [v₁ … v_n] and B = [w₁ … w_n] be mxn matrices. the sum of A and B is…

the mxn matrix A + B := [v₁ + w₁ … v_n + w_n]

let A = [v₁ … v_n] be a mxn matric and c a real number. the scalar product  of A with c is…

the mxn matrix cA := [cv₁ … cv_n]

let A be a mxk matrix and B = [b₁ … b_n] be a kxn matrix. then, the matrix product of A and B is…

the mxn matrix AB = [Ab₁ … Ab_n]

the identity matrix I_n is…

the nxn matrix I_n = [e₁ … e_n]

let A be a nxn matrix. the inverse of A, if it exists, is…

the matrix B such that AB = BA = I_n, where B = A^-1

an nxn matrix is called elementary if:

it can be obtained by performing exactly one operation to the identity matrix

the unit square is the subset of R² given by…

S := {a₁e₁ + a₂e₂ : 0 <= a₁, a₂ <= 1}

an ordered basis {b₁, b₂} for R² is called positively oriented if:

we can rotate b₁ counterclockwise to reach b₂ without crossing the line spanned by b₂

let F: R²→R² be a linear transformation. then, the determinant of F, denoted by det(F) is…

the oriented area of F(S). that is, det(F) :=

1. area(F(S)), if {F(e₁), F(e₂)} is positively oriented

2. -area(F(S)), if {F(e₁), F(e₂)} is negatively oriented

3. 0, if area(F(S)) = 0

if A is a 2×2 matrix, the determinant of A, denoted by det(A) is…

the determinant of the matrix transformation T_A. that is, det(A) := det(T_A)

the unit cube is the subset of R³ given by…

C := {a₁e₁ + a₂e₂ + a₃e₃ : 0 <= a_1, a₂, a₃ <= 1}

for a nxn matrix A = (a_ij), the ij-minor of A is…

the (n-1)x(n-1) matrix A_ij with the i^th row and j^th column deleted

let A be the nxn matrix with ij-entry equal to a_ij. we define the determinant of A by the following cofactor expansion formula:

det(A) := a₁₁det(A₁₁) - a₁₂det(A₁₂) + … + (-1)ⁿ⁺¹a_1ndet(A_1n)

let A be a nxn matrix. a non-zero vector v is an eigenvector of A if:

there exists a real number λ such that Av = λv. the scalar λ is called an eigenvalue of A.

let A be a nxn matrix with eigenvalue λ. the λ-Eigenspace of A is…

the vector subspace of Rⁿ defined by E_λ := Nul(A - λI_n)

let A be a nxn matrix with eigenvalue λ. the geometric multiplicity of λ is…

the dimension of the λ-eigenspace, dim(E_λ)

for a nxn matrix A, the characteristic polynomial of A is…

_XA(x) := det(A - xI_n)

let B = {v₁, …, v_n} be an ordered basis for a vector space V. recall that every vector x in V can be written in the form x = x₁v₁ + … + x_nv_n. the B-coordinates of x is the vector in Rⁿ given by…

[x]_B = [x₁ … x_n]

let C and B be bases for a vector space V. then the change of base matrix M_C←B is the matrix satisfying…

M_C←B[x]_B = [x]_C, for every vector x in V

let C be a basis for a vector space V. then for any vector x and y in V and real number scalar k…

[x + y]_C = [x]_C + [y]_C and [kx]_C = k[x]_C

let {b₁, …, b_n} be a linearly independent subset of a vector space V. then, for any basis C of V, the set {[b₁]_C, …, [b_n]_C} is…

also linearly independent

let F: Rⁿ → Rⁿ be a linear transformation, and B be any basis for Rⁿ. then, the defining matrix of F with respect to the basis B is…

the matrix M so that [F(v)]_B = M[x]_B, where M = M_F,B

two nxn matrices B and C are called similar if:

they represent the same function, but in possibly different bases. that is, there is a single linear function F: Rⁿ → Rⁿ so that  M_F,B = B and M_F,C = C, where B and C are bases for Rⁿ

a matrix D is called diagonal if:

the only non-zero entries in the matrix appear on the diagonal. we write D = diag(λ₁, …, λ_n)

an nxn matrix A is called diagonalizable if:

it is similar to a diagonal matrix

suppose that A is a nxn diagonalizable matrix with eigenvalues λ₁, …, λ_n and corresponding linearly independent eigenvectors v₁, …, v_n. we call the equality

A = CDC^-1 the eigendecomposition of the matrix A

the dot product of u and v is the scalar…

u • v := u₁v₁ + u₂v₂ + … + u_nv_n

let u and v be vectors in Rⁿ. The norm of a vector u in Rⁿ is…

||u|| := √(u • u)

let u and v be vectors in Rⁿ. the distance between vectors u and v is…

d(u, v) := ||u - v||

let u and v be vectors in Rⁿ. we say that u and v are orthogonal if…

u • v = 0

a basis B = {v₁, v₂, …, v_n} is orthogonal if…

v_i • v_j = 0 for every i ≠ j

a basis B is called orthonormal if its orthogonal and…

||v_i|| = 1 for every v_i in B

we call an nxn matrix Q orthogonal if…

its column vectors form an orthonormal basis for Rⁿ

for vectors x, y in Rⁿ, the orthogonal projection of x onto y is…

proj_yx : [(x • y) / (y • y)] * y

an nxn matrix A is called orthogonally diagonalizable if…

there exists an orthogonal matrix Q and diagonal matrix D such that Q^TAQ = D

suppose that A is an nxn symmetric matrix with eigenvalues λ₁, …, λ_n and orthonormal basis of eigenvectors {v₁, …, v_n}. we call the equality

A = QDQ^T a spectral decomposition of A, where D = diag(λ₁, …, λ_n) and Q = [v₁ … v_n]

let A be a mxn matrix and {v₁, …, v_n} be an orthonormal basis for Rⁿ of eigenvectors for A^TA, as above. the singular values of A are…

σ_i := ||Av_i||