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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.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · 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]

ជំហាន​ដោយ​ជំហាន

  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]

បង្ហាញ​ចម្លើយ
A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

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គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

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ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: 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.

សំណួរ​ដែល​មនុស្ស​សួរ

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.

បន្ថែម​ទៀត​ក្នុង Linear Algebra

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