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Vector spaces, span and linear independence

The axioms, spanning sets, independence, basis and dimension.

A vector space is a set closed under addition and scaling. Vectors are independent when no combination cancels; a basis is an independent spanning set, and every basis has the same size: the dimension. Picture it: two vectors spanning a plane unless they are parallel, the 3D view of the column vectors. Think it: rank is the dimension of the column space; rank + nullity = number of columns.

Exemple résolu · rank of [[1,2],[2,4]]

Linear-Algebra · Rank

Rank of [[1,2],[2,4]]

\left[\begin{matrix}1 & 2\\2 & 4\end{matrix}\right]

Étape par étape

  1. \left[\begin{matrix}1 & 2\\2 & 4\end{matrix}\right]

    Row-reduce; the rank is the number of non-zero rows (pivots).

  2. \left[\begin{matrix}1 & 2\\0 & 0\end{matrix}\right]

    R2 ← R2 − (2)·R1 to clear column 1.

  3. \operatorname{rank} A = 1

Révèle la réponse
\operatorname{rank} A = 1

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Comment faire: Vector spaces, span and linear independence

  1. Put the vectors as columns of a matrix and row-reduce.
  2. Pivot columns are a basis of the span; the number of pivots is the rank.
  3. Free columns give the null space: the dependencies among the vectors.

Questions posées par les gens

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

Plus en Linear Algebra

DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis