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Geometric statements in compressed form

A grid of small interactive figures — lattices, subspaces, projections, Gram-Schmidt, rank-nullity, and the SVD ellipse. Each figure isolates one geometric claim. Click to expand a figure to full width for closer reading.

R² / R³ lattice
e\u2081e\u2082
R2={(x1,x2):xi∈R}\mathbb{R}^2 = \{(x_1, x_2) : x_i \in \mathbb{R}\}

The plane as a vector space with canonical basis e₁, e₂.

Subspace through origin
0span(v)
W=span⁡(v)={cv:c∈R}W = \operatorname{span}(\mathbf{v}) = \{c\mathbf{v} : c \in \mathbb{R}\}

A line through the origin is a 1-dimensional subspace of R².

Orthogonal projection
proj\u1D65(x)
proj⁡v(x)=x⋅vv⋅vv\operatorname{proj}_{\mathbf{v}}(\mathbf{x}) = \frac{\mathbf{x}\cdot\mathbf{v}}{\mathbf{v}\cdot\mathbf{v}}\mathbf{v}

Projection of x onto the line span(v); the residual is orthogonal to v.

Gram-Schmidt
step 1: normalize v\u2081 \u2192 u\u2081step 2: v\u2082 - (v\u2082\u00B7u\u2081)u\u2081, normalize
u2=v2−(v2⋅u1)u1∥⋅∥\mathbf{u}_2 = \frac{\mathbf{v}_2 - (\mathbf{v}_2 \cdot \mathbf{u}_1)\mathbf{u}_1}{\|\cdot\|}

Two steps: normalize v₁, then subtract its component from v₂ and normalize.

Rank-nullity
\u211D\u207F\u211D\u1D50Aker(A)im(A)dim ker + dim im = n
dim⁡ker⁡(A)+rank⁡(A)=n\dim\ker(A) + \operatorname{rank}(A) = n

The kernel and image of a linear map partition the domain dimension.

SVD geometric
\u03C3\u2081 = 3\u03C3\u2082 = 1
A=UΣV⊤A = U\Sigma V^\top

A maps the unit circle to an ellipse; singular values σᵢ are the semi-axes.

Click any figure to expand. Each figure is a compressed geometric statement — subspaces, projections, and decompositions.

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Figures Gallery — Linear Algebra Labs · hbar.university