Geometric Bites

Let A, B and X be column vectors in ℝ³: A = a₁ a₂ a₃ , B = b₁ b₂ b₃ , X = x y z . BAᵀX = B(AᵀX). Matrix multiplication is associative, so AᵀX may be evaluated first. Since Aᵀ is a 1 × 3 row matrix and X is a 3 × 1 column matrix, their product is a scalar: AᵀX = a₁x + a₂y + a₃z. Therefore, BAᵀX = b₁ b₂ b₃ (a₁x + a₂y + a₃z). The quantity in parentheses is a scalar, so it multiplies every component of B: BAᵀX = b₁a₁x + b₁a₂y + b₁a₃z b₂a₁x + b₂a₂y + b₂a₃z b₃a₁x + b₃a₂y + b₃a₃z . Collect the coefficients of x, y and z into a matrix multiplying X: BAᵀX = b₁a₁ b₁a₂ b₁a₃ b₂a₁ b₂a₂ b...

Let v = p q r ≠ 0 0 0 The objective is to construct the standard 3 × 3 matrix [v]ₓ associated with v, without assuming its entries in advance. Begin with two requirements: Kᵀ = −K Kv = 0. The first condition requires skew-symmetry. The second requires v to lie in the null space of K. 1. Write the general skew-symmetric matrix Since Kᵀ = −K, the diagonal entries must be zero and entries reflected across the diagonal must have opposite signs. Therefore, K = 0 α β −α 0 γ −β −γ 0 At this stage, α, β and γ are unknown. 2. Require Kv = 0 0 α β −α 0 γ −β −γ 0 p q r ...

A skew-symmetric matrix is a square matrix whose transpose equals its negative. The construction below begins with an arbitrary 3 × 3 matrix, subtracts its transpose, and derives the complete general form without assuming the result in advance. K is skew-symmetric precisely when Kᵀ = −K. 1. Begin with an arbitrary matrix Let A = a b c d e f g h i and Aᵀ = a d g b e h c f i Transposition reflects the entries across the main diagonal: rows become columns and columns become rows. 2. Subtract the transpose Define K = A − Aᵀ. Subtract corresponding entries: K = a−a b−d c−g d−b e−e f−h g−c h−f i−i The diagonal entries cancel: ...

A matrix transforms input vectors into output vectors. Some independent directions survive, some are combined, and some may disappear completely into the zero vector. Span, dimension, column space, rank, null space and nullity describe this process precisely. input space → matrix transformation → column space 1. Span: all reachable combinations Given vectors v₁, v₂, …, vₖ, their span is the set of every vector that can be made by scaling and adding them: Span{v₁, v₂, …, vₖ} = {c₁v₁ + c₂v₂ + ⋯ + cₖvₖ : c₁, c₂, …, cₖ ∈ ℝ}. One non-zero direction spans a line through the origin. Two independent directions span a plane through the origin. Three independent directions in ℝ³ span all of ℝ³. 2. Independence and dimension Vectors are linearly independent when none of them can be constructed from the others. Each independent vector contributes a genuinely new direction. A dependent vector contributes no new...

This proof demonstrates that subtracting the transpose of any real 3 × 3 matrix from the original matrix produces a 3 × 3 skew-symmetric matrix. Construction Let M = ( a b c d e f g h i ) Its transpose is M T = ( a d g b e h c f i ) Subtracting the transpose from the original matrix gives K = M − M T = ( 0 b − d c − g d − b 0 f − h g − c h − f 0 ) ...

The Transpose, Symmetric Matrices, Identity Matrices and Zero Matrices Matrices contain more structure than simple rows and columns. Many important ideas in linear algebra come from operations such as reflecting a matrix, recognising symmetry, and identifying matrices that leave vectors unchanged. This article covers four core ideas: the transpose of a matrix symmetric matrices the identity matrix the zero matrix The Transpose of a Matrix The transpose of a matrix is created by swapping its rows and columns. If a matrix A has an entry in row i, column j, then AT has the same entry in row j, column i. Example: A = [ 1 4 ] [ 2 5 ] [ 3 6 ] Its transpose is: AT = [ 1 2 3 ] [ 4 5 6 ] If A is n × m, then AT is m × n. A square matrix stays square, but its entries reflect across the main diagonal. Symmetric Matrices A square matrix is symmetric when it equals its own transpose: A = AT This means the matrix does not ch...