Solving systems of linear equations, Gaussian elimination, and row reduction from a college linear algebra course.
36 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a linear equation in n variables?
- An equation of the form a1*x1 + a2*x2 + ... + an*xn = b, where a1, a2, ..., an, b are constants and x1, x2, ..., xn are variables.
- What is a system of linear equations?
- A collection of two or more linear equations in the same variables.
- When are two systems of linear equations equivalent?
- When they have exactly the same solution set.
- What is an augmented matrix?
- A matrix formed by appending the constant column to the coefficient matrix.
- What is a coefficient matrix?
- The matrix containing only the coefficients of the variables, without the constants.
- Name the three elementary row operations.
- Swap two rows, multiply a row by a nonzero constant, and add a multiple of one row to another.
- Two matrices are row equivalent if...
- One can be obtained from the other by a sequence of elementary row operations.
- What is row echelon form (REF)?
- A matrix where each leading entry is to the right of the leading entry in the row above, and all entries below a leading entry are zero.
- What is reduced row echelon form (RREF)?
- A matrix in row echelon form where each leading entry is 1 and is the only nonzero entry in its column.
- What is Gaussian elimination?
- A method for solving systems by reducing the augmented matrix to row echelon form using elementary row operations.
- What is Gauss-Jordan elimination?
- Gaussian elimination followed by row operations to reduce the augmented matrix to reduced row echelon form.
- What is a leading entry (pivot)?
- The leftmost nonzero entry in a row.
- What is a free variable?
- A variable that does not correspond to a leading entry in the row echelon form.
- What is back substitution?
- A method for finding solutions by starting from the last row of a row echelon matrix and solving for variables in reverse order.
- When does a system have a unique solution?
- When the system is consistent and every variable is a leading variable (no free variables).