Linear Algebra Concepts - Notes

Chapter 1: Systems of Linear Equations

  • Definition of a Linear Equation:

    • A linear equation in variables $x1, x2, ext{…}, xn$ is expressed as: a</em>1x<em>1+a</em>2x<em>2+⋅⋅⋅+a</em>nxn=ba</em>1x<em>1 + a</em>2x<em>2 + ··· + a</em>nx_n = b
    • Where $a_i$ are real coefficients and $b$ is a constant.
    • A solution to the equation is an n-tuple $(x1, x2, … x_n)$ that satisfies the equation.
    • If $b = 0$, the equation is termed homogeneous.
    • Example: The triplet $(1, 1, 2)$ is a solution of the equation $2x1 - 3x2 + x_3 = 1$.

  • Definition of a System of Linear Equations:

    • A system consists of m linear equations:
      egin{cases}
      a{11}x1 + a{12}x2 + ext{…} + a{1n}xn = b1 \ a{m1}x1 + a{m2}x2 + ext{…} + a{mn}xn = bm \
      ext{Where } a{ij} ext{ are real numbers for } 1 ext{ ≤ i ≤ m, } 1 ext{ ≤ j ≤ n.}\ ext{A solution is a tuple that satisfies all equations.} \ ext{Homogeneous Definition}: If all } bi = 0, <br /> ext{otherwise, it is non-homogeneous.}
      \n<br /> <br /> ext{Property: All homogeneous systems have the trivial solution } (0,0,…,0).
      \
      ext{Set of all solutions is called } solution ext{ set.}
      \
      \
Examples

  • Example 1: For the homogeneous system:
    \egin{align} 2x - y + z & = 0 \ x - 4y + 4z & = 0 \ ext{ Solution Set: }\ ext{Clearly, (0, 0, 0) is a solution.}\ ext{If } E2 ext{ = } 2E1 - E1 ext{ yields } y = z. \ ext{This means the solution set is in the form } (0, z, z) ,\ ext{ with } z ext{ taking any real value.} \end{align}

  • Example 2: Solve the system:
    \egin{align} y - 3z & = 4\ 2z & = 6 \ ext{Solving gives: } z = 3, y = 13, x = -16.\ ext{Therefore, the solution is } (-16, 13, 3). \end{align}

  • Example 3: For:
    \egin{align} (1, 2, 4) ext{ yields solutions through the method shown above.}\ Find all solutions systematically. \, ext{Set the augmented matrix then apply Gaussian elimination.} \end{align}

  • Matrix Representation: The augmented matrix from a linear system describes all equations. Transforming one system into another can be done through permissible operations defined as:

    1. Exchange rows.
    2. Add multiples of rows.
    3. Multiply a row by a non-zero scalar.
Classifications of Linear Systems
  • Types of systems:
    • Incompatible (no solutions) noted as S.I.
    • Compatible determined (one solution) noted as S.C.D.
    • Compatible indeterminate (infinitely many solutions) noted as S.C.I.
Matrix Representation of Linear Systems

  • Definition: An augmented matrix is formed as follows:
    \begin{pmatrix}
    a{11} & a{12} & … & a{1n} & b1\
    a{21} & a{22} & … & a{2n} & b2\
    … & … & … & … & … \
    a{m1} & a{m2} & … & a{mn} & bm
    \end{pmatrix}

  • Example of Matrix Operations:

    • The operations include swapping, scalar multiplication, and adding rows/multiple of a row to another.
Gaussian Elimination Method
  • A method to determine if a system has solutions and to find them: Transform the system to triangular form.
Examples of Gaussian Elimination
  • Example 5: For the system:
    \begin{align} x + 4y + 2z & = -4 \ 3x + 2y - 2z & = 0 \ 2x - y + 3z & = 9 \ ext{Form the augmented matrix and start operations:}\ \end{align}$$