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Math Notes

University Mathematics

ConceptComplete

Inner Product Definition and Properties

An inner product generalizes the dot product to abstract vector spaces, introducing notions of length, angle, and orthogonality. This structure connects algebra to geometry.

DefinitionInner Product

An inner product on a real vector space VV is a function ⟨⋅,⋅⟩:V×V→R\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R} satisfying for all u,v,w∈V\mathbf{u}, \mathbf{v}, \mathbf{w} \in V and c∈Rc \in \mathbb{R}:

  1. Positivity: ⟨v,v⟩≥0\langle \mathbf{v}, \mathbf{v} \rangle \geq 0
  2. Definiteness: ⟨v,v⟩=0\langle \mathbf{v}, \mathbf{v} \rangle = 0 if and only if v=0\mathbf{v} = \mathbf{0}
  3. Symmetry: ⟨u,v⟩=⟨v,u⟩\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle
  4. Linearity in first argument: ⟨cu+v,w⟩=c⟨u,w⟩+⟨v,w⟩\langle c\mathbf{u} + \mathbf{v}, \mathbf{w} \rangle = c\langle \mathbf{u}, \mathbf{w} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle

A vector space with an inner product is called an inner product space.

For complex vector spaces, symmetry is replaced by conjugate symmetry: ⟨u,v⟩=⟨v,u⟩‾\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}.

ExampleStandard Inner Products

  1. Euclidean inner product on Rn\mathbb{R}^n: ⟨x,y⟩=x⋅y=∑i=1nxiyi\langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^n x_iy_i

  2. Function space C[a,b]C[a,b]: ⟨f,g⟩=∫abf(x)g(x) dx\langle f, g \rangle = \int_a^b f(x)g(x)\,dx

  3. Weighted inner product: ⟨x,y⟩w=∑i=1nwixiyi\langle \mathbf{x}, \mathbf{y} \rangle_w = \sum_{i=1}^n w_ix_iy_i where wi>0w_i > 0

  4. Matrix space Mn(R)M_n(\mathbb{R}): ⟨A,B⟩=tr(ATB)=∑i,jaijbij\langle A, B \rangle = \text{tr}(A^TB) = \sum_{i,j} a_{ij}b_{ij} (Frobenius inner product)

DefinitionNorm and Distance

The norm (or length) induced by an inner product is: ∥v∥=⟨v,v⟩\|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}

The distance between vectors is: d(u,v)=∥u−v∥=⟨u−v,u−v⟩d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\| = \sqrt{\langle \mathbf{u} - \mathbf{v}, \mathbf{u} - \mathbf{v} \rangle}

A vector with ∥v∥=1\|\mathbf{v}\| = 1 is called a unit vector. The process normalization produces a unit vector: v^=v∥v∥\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}.

DefinitionAngle and Orthogonality

For nonzero vectors u,v\mathbf{u}, \mathbf{v}, the angle θ\theta between them satisfies: cos⁡θ=⟨u,v⟩∥u∥∥v∥\cos\theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\|\|\mathbf{v}\|}

Vectors u\mathbf{u} and v\mathbf{v} are orthogonal (denoted u⊥v\mathbf{u} \perp \mathbf{v}) if ⟨u,v⟩=0\langle \mathbf{u}, \mathbf{v} \rangle = 0.

Remark

The inner product axioms capture the essential properties that make geometric intuition work: positivity ensures "length" is non-negative, definiteness says only the zero vector has zero length, symmetry makes the product commutative, and linearity allows algebraic manipulation. These properties enable generalization of Euclidean geometry to infinite-dimensional spaces like function spaces.