maths.free › Linear Algebra › Orthogonality, projections and least squares

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.

उदाहरण · 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]

चरणद्वारा चरण

  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]

जवाफ प्रकट गर्नुहोस्
A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।

तपाईँको आफ्नै काम गर्ने राख्नुहोस्

एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.

साइन अप गर्नुहोस् लगइन

यहाँ प्रयोग गरिएको प्रतीक

पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।

कसरी: 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.

प्रश्नहरू मानिसहरूले सोध्छन्

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.

यसमा थप Linear Algebra

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