Linear Algebra Questions and Answers 2023.
Define a Linear equation
Correct Answers: An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2,
... [Show More] 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 [Show Less]