Linear Algebra Guide
Linear equation - ANSWER>>>An equation that can be written as a1x1 +
a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance
consistent system - ANSWER>>>Has one or infinitely many solutions
inconsistent system - ANSWER>>>Has no solution
leading entry - ANSWER>>>Leftmost non-zero entry in a non-zero row
Echelon form - ANSWER>>>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 - ANSWER>>>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 - ANSWER>>>the collection of all vectors in R^n that can be written
as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)
Ax = b - ANSWER>>>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 - ANSWER>>>A position in the original matrix that
corresponds to a leading 1 in a reduced echelon matrix
pivot column - ANSWER>>>A column that contains a pivot position
homogeneous - ANSWER>>>A system that can be written as Ax = 0; the x
= 0 solution is a TRIVIAL solution
independent - ANSWER>>>If only the trivial solution exists for a linear
equation; the columns of A are independent if only the trivial solution exists
Linear equation - ANSWER>>>An equation that can be written as a1x1 +
a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance
consistent system - ANSWER>>>Has one or infinitely many solutions
inconsistent system - ANSWER>>>Has no solution
leading entry - ANSWER>>>Leftmost non-zero entry in a non-zero row
Echelon form - ANSWER>>>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 - ANSWER>>>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 - ANSWER>>>the collection of all vectors in R^n that can be written
as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)
Ax = b - ANSWER>>>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 - ANSWER>>>A position in the original matrix that
corresponds to a leading 1 in a reduced echelon matrix
pivot column - ANSWER>>>A column that contains a pivot position
homogeneous - ANSWER>>>A system that can be written as Ax = 0; the x
= 0 solution is a TRIVIAL solution
independent - ANSWER>>>If only the trivial solution exists for a linear
equation; the columns of A are independent if only the trivial solution exists
