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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.

Esempio di funzionamento: 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]

Passo dopo passo

  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]

Rivela la risposta
A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

Symbols used here

\det A,\ |A|
determinant
Scaling factor of area/volume under A; zero means singular.
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
matrix
A rectangular array of numbers; a linear map.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
A^{-1},\ A^{T}
inverse, transpose
The matrix that undoes A; A with rows and columns swapped.
\lambda
lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).

How to: 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.

Questions people ask

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.

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Più in Linear Algebra

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