02 - Elementary Row Operations (EROs)

Elementary Row Operations (given a linear system)

Brackets is when applying it to a matrix (e.g. an Augmented Matrix)

Swapping the order of two equations (or rows)

Multiplying an equation (or row) by a non-zero scalar

Adding a multiple of one equation (or row) to another equation (or row)

Notation when applying it to matrices (or equations)

Row Swap: Swap two rows ( $R_{i} \leftrightarrow R_{j}$ )
→ Notation: $S_{ij}$

Row Scaling: Multiply a row by a nonzero scalar ( $λ$ )
→ Notation: $M_{i} (λ)$ , where ( $R_{i} \to λ R_{i}$ )

Row Addition: Add a multiple of one row to another
→ Notation: $A_{ij} (λ)$ , where ( $R_{j} \to R_{j} + λ R_{i}$ )

Not standard notation but convenient to use, $R_{i}$ refers to the $i^{t h}$ row

Using EROs doesn't affect the solution space of Linear Systems

Elementary Matrices

$the (i, j) th entry of S_{I J} the (i, j) th entry of M_{I} (λ) the (i, j) th entry of A_{I J} (λ) = ⎩ ⎨ ⎧ 1110 if i = j \neq = I, J if i = J, j = I if i = I, j = J otherwise = ⎩ ⎨ ⎧ 1 λ 0 if i = j \neq = I if i = j = I otherwise = ⎩ ⎨ ⎧ 1 λ 0 if i = j if i = J, j = I otherwise$
These elementary matrices can also be obtained by applying the respective ERO onto the identity matrix!

Inverses of the Elementary Matrices

$(S_{ij})^{- 1} = S_{ji} = S_{ij}; (A_{ij} (λ))^{- 1} = A_{ij} (- λ); (M_{i} (λ))^{- 1} = M_{i} (λ^{- 1})$

Invariance of Solution Space under EROs corollary

Let $(A ∣ b)$ be a linear system of $m$ equations and $E$ an $m \times m$ elementary matrix
Then
$x is a solution of (A ∣ b) ⟺ x is a solution of (E A ∣ E b)$

Proof

As $E$ is invertible then (by pre-multiplication of $E$ )
$A x = b ⟹ E A x = E b$
And (by pre-multiplication of $E^{- 1}$ )
$E A x = E b ⟹ A x = b$

Oxford Maths Notes

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02 - Elementary Row Operations (EROs)

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