Linear Algebra: Matrix Transformations and Multiplications

Category: Linear Algebra & Matrix Calculus

1. Matrices as Spatial Operations

A matrix represents a linear spatial transformation map. Multiplying a matrix by a vector rotates, stretches, or shears coordinate positions within an n-dimensional space.

2. The Core Mechanic of Matrix Multiplication

To multiply two matrices together, the inner dimensions must match perfectly. If Matrix A is size m × n and Matrix B is size n × p, the output Matrix C will be size m × p. You multiply rows by columns using dot product summation tracking.

3. Proof of Non-Commutativity

In standard arithmetic, scalar multiplication is commutative: 5 × 3 = 3 × 5. In linear algebra, matrix operations are strictly non-commutative. Let us prove mathematically that A × B ≠ B × A using an explicit algebraic counterexample matrix map.

Let Matrix A = [[1, 2], [3, 4]] and Matrix B = [[0, 1], [2, 3]]

Calculate Product AB (Row × Column):
Row 1: [(1*0 + 2*2), (1*1 + 2*3)] = [4, 7]
Row 2: [(3*0 + 4*2), (3*1 + 4*3)] = [8, 15]
AB = [[4, 7], [8, 15]]

Calculate Product BA (Row × Column):
Row 1: [(0*1 + 1*3), (0*2 + 1*4)] = [3, 4]
Row 2: [(2*1 + 3*3), (2*2 + 3*4)] = [11, 16]
BA = [[3, 4], [11, 16]]

Conclusion: Since AB ≠ BA, matrix multiplication order dictates structural output paths.

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