Linear Algebra

Linear Algebra Questions and Answers 2023. Define a Linear equation Correct Answers: An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance Define a Consistent system Correct Answers: Has one or infinitely many solutions Define an Inconsistent system Correct Answers: Has no solution Define a leading entry Correct Answers: Leftmost non-zero entry in a non-zero row Define an Echelon form Correct Answers: 1. All nonzero rows are above 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 Define a Reduced Echelon Form Correct Answers: 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 Define a Span Correct Answers: the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants) Work out Ax = b Correct Answers: 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 Define a Pivot position Correct Answers: A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix Define a Pivot column Correct Answers: A column that contains a pivot position. Homogeneous Correct Answers: A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution. Independent Correct Answers: If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists. Dependent Correct Answers: If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries Transformation Correct Answers: assigns each vector x in R^n a vector T(x) in R^m Matrix multiplication warnings Correct Answers: 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 Correct Answers: flips rows and columns Properties of transposition Correct Answers: 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 Invertibility rules Correct Answers: 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) Correct Answers: 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