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First steps

Vectors as coordinates, matrices as instructions and the first structural ideas

How do coordinates, matrices and equations fit into one picture?

This chapter introduces the basic objects of linear algebra and connects them immediately to structure. Vectors are coordinates, matrices are instructions, matrix multiplication becomes transformation and the same viewpoint leads to span, independence, orthogonality, transpose and the first encounter with A x = b.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

The chapter starts with notation, but the deeper point is that the same few structures reappear everywhere.

  • A vector is a list of coordinates and a matrix tells how each coordinate direction moves
  • Matrix-vector multiplication is built from scaled columns, so the columns of a matrix already encode its action
  • Linearity means addition and scaling are preserved, which is why grids become skewed but still structured
  • Span and independence describe whether vectors create new directions or repeat old ones
  • Dot product and orthogonality measure directional overlap and prepare the way for projection
  • Transpose changes rows into columns and begins the bridge between matrix domain, codomain and linear equations

This chapter is a map of the ideas that later chapters develop in a more specialized way.

Why does matrix-vector multiplication sit at the center of the chapter?

Because it connects coordinates, columns, transformations and equations in one definition. Once you see a matrix as acting on vectors by combining its columns, many later ideas become variations of the same structure.

Related chapters

  • Matrix multiplication develops the structure of matrix products beyond the introductory viewpoint given here
  • Linear equations continues from the first encounter with Ax = b and develops a full algorithmic treatment
  • Linear transformations expands the idea of matrices as instructions into a full geometric language of maps between spaces
  • Projection builds on dot product and orthogonality to separate vectors into kept and discarded components
  • The four fundamental subspaces develops span, transpose and equations into a structural picture of domain, codomain and orthogonal decomposition

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Space, vector, matrix 1–2
Matrix-vector multiplication 2–3
Linearity of matrix-vector multiplication 3–5
Matrix-vector multiplication ℝ² → ℝ³ 5–6
Matrix-matrix multiplication 6–8
Concept summary 9
Linear independence & linear dependence 10–11
Vector span & basis of a space 11–12
Matrix multiplication vs matrix addition 12–13
Dot product ℝⁿ × ℝⁿ → ℝ 13–14
Geometry of dot product 14–16
Orthogonality 16–17
Matrix transpose 17–20
Linear equations, first encounter 20–21
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Why do span and independence appear so early?

Because they answer the first structural question about a set of vectors: do these vectors create new directions or not. That question later controls basis, rank, solvability and the geometry of matrix action.

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