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Dual Perspectives on Linear Systems
MATH004 Lesson 2
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The foundation of linear algebra rests on two distinct yet mathematically equivalent interpretations of the equation $Ax = b$. We transition from the traditional Row Picture, where we seek the intersection of geometric hyperplanes, to the more powerful Column Picture, which views the matrix $A$ as a set of basis vectors linearly combined to construct the target vector $b$.

1. The Geometry of the Solution

In the Row Perspective, each equation in a 3x3 system represents a plane in $\mathbb{R}^3$. The solution $x = (2, 3, 4)$ is the unique point where these three planes intersect. Mathematically, $b$ is calculated one row at a time using the inner product (a row times a column):

$b = [A(1, :) * x; A(2, :) * x; A(3, :) * x]$

Conversely, the Column Picture interprets $Ax = b$ as a request for a specific linear combination of column vectors: $b = A(:, 1)x_1 + A(:, 2)x_2 + A(:, 3)x_3$. Here, the matrix $A$ is seen as a collection of directions, and the variables $x_i$ are the weights (scalars) assigned to reach the destination $b$. As highlighted in the core theory: Column picture: $Ax = b$ asks for a combination of columns to produce $b$.

Worked Example 2.1 A

Consider $A = \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix}$. Calculating $ad - bc$ gives $2 - 2 = 0$. This matrix is singular. In the row picture, the lines are parallel. In the column picture, both columns lie on the same line; we cannot reach a $b$ that is not on that line.

2. A as a Linear Transformation

Multiplying a vector by $A$ is not just a calculation; it is a linear transformation. It satisfies the principle of linearity: $Aw = cAu + dAv$ (where $w = cu + dv$). This confirms $A$ is an operator that maps vectors from one space to another, potentially involving rotation or projection (Diagram, pg 42).

  • Dimension Rule: $(m \times n)(n \times p) = (m \times p)$ (Page 72).
  • Identity Components: Standard basis vectors $e_1 = [1,0,0]^T, e_2 = [0,1,0]^T, e_3 = [0,0,1]^T$ define the dimensions of this space (Diagram, pg 80).
  • Advanced Note: The Woodbury-Morrison formula is the 'matrix inversion lemma' in engineering, used for updating inverses after small changes to $A$.
🎯 Core Principle
$Ax = b$ is solved by finding how much of each column vector ($x_n$) we need to combine to hit the target $b$. If $A$ is invertible, the one and only solution is $x = A^{-1}b$.