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Media Library

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Summarize

A B U A △ B

Venn diagram of the symmetric difference \(A \triangle B\).

Sets

A B U (A ∪ B) = A ∩ B

Venn diagram of De Morgan's law \((A \cup B)^c = A^c \cap B^c\).

Sets

x y 1 v=(0.5, 1) u=(1,2) u+v=(1.5, 3) 1 2 2 3 3 W : y = 2x All vectors lie on the line y = 2x, illustrating closure under addition and scalar multiplication.

A subspace of \(\mathbb{R}^2\) consisting of all vectors lying on the line \(y = 2x\), closed under vector addition and scalar multiplication.

Vector Spaces

Cayley graph of D₆ generated by r and s. Solid arrows show multiplication by r; dashed edges show multiplication by s. e r r² r³ s rs r²s r³s r⁴s r⁵s multiply by r multiply by s r⁴ r⁵

Cayley graph of the dihedral group \(D_6\) for the generating set \(\{r, s\}\), with arrows recording multiplication by \(r\) and by \(s\).

Dihedral Groups

Every vector u splits into a multiple of v and a remainder w orthogonal to v. The foot of the perpendicular is the point cv. O v u w cv

Orthogonal projection of a vector \(\mathbf{u}\) onto the line spanned by \(\mathbf{v}\), with the decomposition \(\mathbf{u} = c\mathbf{v} + \mathbf{w}\) where \(\mathbf{w}\) is orthogonal to \(\mathbf{v}\) and \(c\mathbf{v}\) is the foot of the perpendicular.

Inner Product Spaces

The Cauchy-Schwarz inequality keeps the ratio of the inner product to the two norms inside the interval from −1 to 1, so the angle θ exists. O u v θ

The angle \(\theta\) between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\), defined through \(\cos\theta = \langle\mathbf{u},\mathbf{v}\rangle/(\|\mathbf{u}\|\|\mathbf{v}\|)\), whose existence is guaranteed by the Cauchy-Schwarz inequality.

Inner Product Spaces

In a parallelogram the squares of the two diagonals add up to the squares of the four sides. A norm with this property is induced by an inner product. O u v u + v u − v

Geometric reading of the parallelogram law \(\|\mathbf{u}+\mathbf{v}\|^2 + \|\mathbf{u}-\mathbf{v}\|^2 = 2\|\mathbf{u}\|^2 + 2\|\mathbf{v}\|^2\), where the squares of the two diagonals add up to the squares of the four sides.

Inner Product Spaces

Removing from v₂ its projection on v₁ leaves w₂ orthogonal to v₁. Dividing each of the two by its norm produces the orthonormal pair e₁, e₂. O v₁ v₂ w₂ e₁ e₂

One step of the Gram-Schmidt process: removing from \(\mathbf{v}_2\) its projection onto \(\mathbf{v}_1\) leaves the orthogonal remainder \(\mathbf{w}_2\), and normalising both vectors produces the orthonormal pair \(\mathbf{e}_1, \mathbf{e}_2\).

Inner Product Spaces

0 1 2 3 -1 -2 1/3 π Rational Natural −√3 Irrational Irrational The real number line illustrating the distribution of natural numbers, integers, rational numbers, and irrational numbers.

Irrational radicals on the real number line, with values such as \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) positioned among rational and natural numbers.

Radicals

A D C B a √a 1 1+a Geometric construction of √a: extend a segment by 1 unit, draw a semicircle, and obtain √a from the perpendicular segment.

The classical compass-and-straightedge construction of the segment of length \(\sqrt{a}\), based on Euclid's theorem and using a semicircle with a perpendicular segment.

Radicals