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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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]

Àwọn ìṣàmúlò-ètò

\det A = 1
A matrix is invertible only when its determinant is non-zero.
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.
A^{-1} = \left[\begin{matrix}1 & -1\\-1 & 2\end{matrix}\right]

Fi àwọn àgbèwọlé hàn

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

Nígbà yìí Yan àwọn àkóónú kan, tàbí tẹ̀ àtí kọ́ àwọn àkóónú rẹ̀. Àwọn ìṣàmúlò-ètò gbogbo, àwòrán, àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: Orthogonality, projections and least squares

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Linear Algebra

Determinants Matrix inverse Eigenvalues and eigenvectors Matrix multiplication Row reduction Vector spaces, span and linear independence Diagonalisation and matrix powers Linear transformations and change of basis