Linear Algebra 2023 with verified questions and answers

linear equation An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance consistent system Has one or infinitely many solutions inconsistent system Has no solution leading entry Leftmost non-zero entry in a non-zero row 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 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 Span the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants) 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 pivot position A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix pivot column A column that contains a pivot position homogeneous A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution independent If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists dependent If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries transformation assigns each vector x in R^n a vector T(x) in R^m 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 Transposition flips rows and columns Properties of transposition 1. (A^T)^T = A; 2. (A+B)^T = A^T + B^T; 3. (rA)^T = rA^T; 4. (AB)^T = B^TA^T 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 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 Column Row Expansion of AB col1Arow1B + ... 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 Leontief input-output model x = Cx + d 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) Column space Set of all the linear combinations of the columns of A Null space Set of all solution to Ax = 0 Basis A linearly independent set in H that spans H; the pivot columns of A form a basis for A's column space Dimension The number of vectors in any basis of H; the zero subspace's dimension is 0 rank The dimension of the column space 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 onto consistent for any b; pivots in all rows 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 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 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 orthonormal An orthogonal set of unit vectors