Linear algebra

Systems of Linear Equations

A system of equations can have:

• No solution (inconsistent)
• One solution (consistent)
• Infinitely many solutions (consistent)

A system is called consistent if it has at least one solution and inconsistent if it has none.

Augmented Matrix Representation

The system:
\(\begin{aligned} a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1n}x_{n} &= b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2n}x_{n} &= b_{2} \\ \vdots \quad &\vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \dots + a_{mn}x_{n} &= b_{m} \end{aligned}\)

can be represented as: \(\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix} = \begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \end{bmatrix}\)

Elementary Row Operations

1. Multiply a row by a non-zero constant.
2. Swap two rows.
3. Add or subtract a multiple of one row to another row.

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Echelon and Reduced Row Echelon Forms

1. Echelon Form:
   \(\begin{bmatrix} p_{11} & p_{12} & \dots & p_{1n} \\ 0 & p_{22} & \dots & p_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & p_{mn} \end{bmatrix}\)

2. Reduced Row Echelon Form (RREF):
   \(\begin{bmatrix} 1 & 0 & \dots & 0 & r_{1} \\ 0 & 1 & \dots & 0 & r_{2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & r_{m} \end{bmatrix}\)

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Homogeneous Linear Systems

The system: \(\begin{aligned} a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1n}x_{n} &= 0 \\ a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2n}x_{n} &= 0 \\ \vdots \quad &\vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \dots &+ a_{mn}x_{n} = 0 \end{aligned}\)

Homogeneous systems always have at least the trivial solution:
\([0, 0, \dots, 0]\)

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Properties of Matrices

1. ( A + B = B + A )
2. ( A + (B + C) = (A + B) + C )
3. ( A(BC) = (AB)C ) (order matters!)
4. ( A(B + C) = AB + AC )
5. ( (A + B)C = AC + BC )
6. ( a(B + C) = aB + aC )
7. ( (a + b)A = aA + bA )
8. ( a(bC) = (ab)C )
9. ( a(BC) = (aB)C )

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Invertibility of a Matrix

For a ( 2 \times 2 ) matrix:
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

The inverse exists if (\det(A) \neq 0):
\(A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)

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Determinants

• Minor: A square matrix obtained by deleting one row and one column from the original matrix.
• Cofactor:
  \(C_{ij} = (-1)^{i+j} M_{ij}\)
  where ( M_{ij} ) is the minor corresponding to element ( a_{ij} ).

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LU Decomposition

A matrix ( A ) can be decomposed as:
\(A = LU\)

where ( L ) is a lower triangular matrix and ( U ) is an upper triangular matrix:

\[L = \begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \end{bmatrix}, \quad U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}\]

Steps:

1. Solve ( LY = B ).
2. Solve ( UX = Y ).