Linear Algebra

Properties of transposition

1. (A^T)^T = A; 2. (A+B)^T = A^T + B^T; 3. (rA)^T = r*A^T; 4. (AB)^T = B^T*A^T

Matrix multiplication warnings

1. AB != BA ; 2. If AB = AC, B does not necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0

Echelon form

1. All nonzero rows are above any all zero rows; 2. Each leading entry is in a column to the right of the previous leading entry; 3. All entries below a leading entry in its column are zeros

Ax = b

1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row

Invertibility rules

1. If A is invertible, (A^-1)^-1 = A; 2. (AB)^-1 = B^-1 * A^-1; 3. (A^T)^-1 = (A^-1)^T

LU Factorization

1. Ly = b; Ux = y; 2. Reduce A to echelon form; 3. Place values in L that, by the same steps, would reduce it to I

Subspaces

1. The zero vector is in H; 2. For u and v in H, u + v is also in H; 3. For u in H, cu is also in H (c is a constant)

orthogonal component

1. x is in W' if x is perpendicular to every vector that spans W; 2. W' is a subspace of R^n

pivot column

A column that contains a pivot position

Invertible Matrix Theorem (either all of them are true or all are false)

A is invertible; A is row equivalent to I; A has n pivot columns; Ax = 0 has only the trivial solution; The columns of A for a linearly independent set; The transformation x --> Ax is one to one; Ax = b has at least one solution for each b in R^n; The columns of A span R^n; x --> Ax maps R^n onto each R^m; there is an n x n matrix C such that CA = I; there is a matrix such that AD = I; A^T is invertible; The columns of A form a basis of R^n; Col A = R^n; dim Col A = n; rank A = n; Nul A = [0]; dim Nul A = 0

Basis

A linearly independent set in H that spans H; the pivot columns of A form a basis for A's column space

pivot position

A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix

orthogonal set

A set of vectors where Ui . Uj = 0 (and i != j); if S is an orthogonal set, S is linearly independent and a basis of the subspace spanned by S

homogeneous

A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution

one-to-one

A transformation that assigns a vector y in R^m for each x in R^n; there's a pivot in every column

linear equation

An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance

orthonormal

An orthogonal set of unit vectors

inconsistent system

Has no solution

consistent system

Has one or infinitely many solutions

dependent

If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries

independent

If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists

leading entry

Leftmost non-zero entry in a non-zero row

Reduced Echelon Form

Same as echelon form, except all leading entries are 1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix

Null space

Set of all solution to Ax = 0

Column space

Set of all the linear combinations of the columns of A

rank

The dimension of the column space

Dimension

The number of vectors in any basis of H; the zero subspace's dimension is 0

inner product

a matrix product u^Tv or u . v where u and v are vectors; if U . V = 0, u and v are orthogonal

transformation

assigns each vector x in R^n a vector T(x) in R^m

Column Row Expansion of AB

col1Arow1B + ...

onto

consistent for any b; pivots in all rows

Transposition

flips rows and columns

Span

the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)

Leontief input-output model

x = Cx + d