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Orthogonality, projections and least squares

Dot products, orthonormal bases, projection onto a subspace, fitting a line.

The projection of b onto the column space of A is the closest point; the least-squares solution solves AᵀA x = Aᵀ b. Gram-Schmidt turns any basis orthonormal. Picture it: dropping a perpendicular from a point onto a plane. Think it: least squares is regression; the normal equations are calculus's "set the gradient to zero" done with matrices.

Fungerat exempel · inverse of [[2,1],[1,1]]

Linear-Algebra · Matrix-Inverse

Inverse of [[2,1],[1,1]]

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

Steg för steg

  1. \det A = 1

    A matrix is invertible only when its determinant is non-zero.

  2. A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{1}\left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

    For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.

  3. A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

Avslöja svaret
A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

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Hur man: Orthogonality, projections and least squares

  1. A matrix is invertible only when its determinant is non-zero.
  2. For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.

Frågor folk frågar

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.

Mer information Linear Algebra

DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceDiagonalisation and matrix powersLinear transformations and change of basis