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Systems and Gaussian elimination · Linear Algebra | Math Infinitum

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Linear Algebra
Systems and Gaussian elimination

Linear Algebra · Chapter 06

Systems and Gaussian elimination

Solving any linear system by hand: the augmented matrix, echelon forms and pivots, free variables, the rank, and inverses found by row reduction.

Solving any linear system by hand: the augmented matrix, echelon forms and pivots, free variables, the rank, and inverses found by row reduction.

A x = b

R_{i} \leftarrow R_{i} + c R_{j}

[A ∣ b]

rank (A)

[A ∣ I] ⟶ [I ∣ A^{- 1}]

R_{i} \leftrightarrow R_{j}

rref (A)

x = x_{p} + x_{h}

Lessons

Build the foundation

01Systems in matrix formA system of linear equations can be written as equations, as one vector equation, or as one matrix equation. For solving by hand the most useful form keeps only the numbers: the augmented matrix, one row per equation and one column per unknown.Read lesson →
02Row echelon formElimination aims at a staircase. Each row starts further right than the one above it, and below each step there are only zeros. A system in that shape is solved from the bottom up, one unknown at a time.Read lesson →
03Free variablesWhen a column has no pivot, its unknown is free: it can take any value, and the other unknowns follow from it. Each free unknown is a direction along which the solutions stretch, so one gives a line of solutions and two give a plane.Read lesson →
04Reduced row echelon formElimination can go further than a staircase. Scale every pivot to 1 and clear the entries above each pivot as well as below it, and the solutions can be read straight off the matrix, with no back substitution.Read lesson →
05Rank and the number of solutionsThe number of pivots is the single number that says most about a matrix, and it is called the rank. Compared with the number of unknowns and with the rank of the augmented matrix, it decides whether a system has no solution, exactly one, or infinitely many.Read lesson →

06Inverting by row reductionTo invert a matrix, write the identity beside it and row-reduce the pair. When the left half has become the identity, the right half has become the inverse. If the left half cannot become the identity, there is no inverse.Read lesson →

07Chapter practiceBring the chapter together: set up an augmented matrix, reduce it, read the pivots and the free variables, decide how many solutions there are, and invert by row reduction. No worked example sits above the answers.Read lesson →