Solving systems of linear equations, Gaussian elimination, and row reduction from a college linear algebra course.
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- What is a linear equation in n variables?
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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?
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A collection of two or more linear equations in the same variables.- When are two systems of linear equations equivalent?
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When they have exactly the same solution set.- What is an augmented matrix?
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A matrix formed by appending the constant column to the coefficient matrix.- What is a coefficient matrix?
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The matrix containing only the coefficients of the variables, without the constants.- Name the three elementary row operations.
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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...
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One can be obtained from the other by a sequence of elementary row operations.- What is row echelon form (REF)?
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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)?
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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?
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A method for solving systems by reducing the augmented matrix to row echelon form using elementary row operations.- What is Gauss-Jordan elimination?
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Gaussian elimination followed by row operations to reduce the augmented matrix to reduced row echelon form.- What is a leading entry (pivot)?
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The leftmost nonzero entry in a row.- What is a free variable?
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A variable that does not correspond to a leading entry in the row echelon form.- What is back substitution?
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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?
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When the system is consistent and every variable is a leading variable (no free variables).
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