(15) Matrix Equation

Matrix Equation: Ax = b

  • The matrix equation is of the form ( Ax = b ), where:

    • ( A ) is an ( m \times n ) matrix.

    • ( x ) is a vector in ( \mathbb{R}^n ).

    • ( b ) is a vector in ( \mathbb{R}^m ).

  • Understanding the equation involves recognizing that:

    • The product ( Ax ) represents a linear combination of the columns of ( A ).

Linear Combinations and Matrix-Vector Multiplication

  • Given the columns ( a_1, a_2, , \ldots, a_n ) of matrix ( A ):

    • The product ( Ax ) is expressed as:

      • ( Ax = x_1 a_1 + x_2 a_2 + \ldots + x_n a_n )

  • The operation ( Ax ) is defined only when the number of columns in ( A ) equals the number of entries in ( x ).

Example Calculations

  1. Example 1:

    • If matrix ( A ) has three columns and vector ( x ) has three entries, ( Ax ) can be computed.

    • Example result: ( A \times x ) gives the matrix ( \begin{pmatrix} 3 & 1 \ -4 & 0 \end{pmatrix} ).

  2. Linear Combination:

    • Linear combination ( -2v_1 + 4v_2 - 8v_3 ) can be represented using matrix ( A = \begin{pmatrix} -2 & 4 & -8 \end{pmatrix} ) and vector ( v = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix} ).

Matrix Equation vs. Vector Equation

  • A vector equation can be expressed as:

    • ( x_1 \begin{pmatrix} 1 \ -2 \end{pmatrix} + x_2 \begin{pmatrix} 2 \ 5 \end{pmatrix} = \begin{pmatrix} 7 \ 4 \end{pmatrix} )

  • The corresponding matrix equation:

    • ( Ax = b ) with ( A ) representing coefficients and ( b ) as the solution vector.

Theorem: Equivalence of Representations

  • The matrix equation ( Ax = b ):

    • Has the same solution set as the vector equation.

    • Both are equivalent to the system of linear equations represented in augmented matrix form.

Different Forms of a System of Linear Equations

  1. Matrix Equation: ( Ax = b )

  2. Vector Equation: ( x_1 a_1 + x_2 a_2 + \ldots = b )

  3. System of Linear Equations:

    • Looks something like this:

      • ( a_1 x_1 + a_2 x_2 = b_1 )

      • ( a_3 x_1 + a_4 x_2 = b_2 )

Consistency of Ax = b

  • To determine consistency:

    • Set up an augmented matrix and perform row operations.

    • Augmented matrix reveals consistency:

      • ( 2b_1 - b_3 = 0 ): consistent when this holds true.

Existence of Solutions

  • The equation ( Ax = b ) has a solution if and only if ( b ) is a linear combination of the columns of ( A ).

  • A key question is whether ( b ) lies within the span of columns of ( A ).

Theorems on Span and Solutions

  • If the columns of ( A ) span ( \mathbb{R}^m ), every ( b ) in ( \mathbb{R}^m ) can be expressed as a linear combination of those columns.

  • The statements:

    1. For each ( b \in \mathbb{R}^m ), the equation ( Ax = b ) has a solution.

    2. ( b ) is a linear combination of the columns of ( A ).

    3. The columns of ( A ) span ( \mathbb{R}^m ).

    4. ( A ) has a pivot in every row.

  • All statements are logically equivalent.

Determining if Columns Span R^4

  • Examine whether columns span by reducing or evaluating coefficients in the matrix:

    • Check for pivot positions in every row.

Matrix-Vector Product Properties

  • To compute ( Ax ):

    • The ( i )-th entry of ( Ax ) is the sum of products from row ( i ) of ( A ) and vector ( x ).

Identity Matrix

  • The identity matrix ( I_n ) satisfies ( I_n x = x ).

    • Examples:

      • ( 2 \times 2 ): ( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} )

      • ( 3 \times 3 ): Similar concept with ones on the diagonal and zeros elsewhere.

Conclusion & Practice Examples

  • Encouraged to practice with examples to apply learned concepts.

  • Solving systems involves setting up augmented matrices and applying row reduction techniques.