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A linear map from a metrizable topological vector space is continuous exactly when it maps every sequence converging to zero to a sequence converging to zero. If continuity fails, a decreasing countable neighborhood basis supplies a null sequence whose images stay outside one fixed neighborhood of zero.

A horizontal shear fixes the horizontal axis pointwise and sends $(x,y)$ to $(x+ky,y)$. Its determinant is one and both eigenvalues are one. A nonzero shear preserves area but generally changes lengths and angles. Composing a shear with a planar rotation can produce a symmetric stretch; having determinant one alone does not make a linear map a shear.

A linear map is injective on the class of sparse vectors of order $s$ exactly when its null space contains no nonzero vector with at most $2s$ nonzero entries. Differences of two sparse vectors have at most $2s$ active coordinates. Conversely, splitting the support of a vector of a $2s$-sparse null vector into two parts constructs two distinct $s$-sparse vectors with the same image. The strict null space property implies sparse injectivity by applying it to both parts of such a split.

The image of the linear map $A+B$ is contained in the sum of the images of $A$ and $B$. Since the dimension of a vector space of this sum is at most the sum of their dimensions, matrix rank is subadditive. Iteration shows that a sum of $r$ matrices of matrix rank at most one has matrix rank at most $r$, over any field.

A tangent vector is a linear map on germs of smooth functions satisfying $D(ab)=a(x)D(b)+b(x)D(a)$. In a manifold chart, first-order expansion shows $Dh={\sum }_{i}D(u^i){\partial }_{i}h(x)$. Thus the coordinate derivations form a basis, and the dimension equals the manifold dimension. Composition of germs with a smooth map defines its differential of a smooth map without choosing charts.

The tensor product $V\otimes W$ of vector spaces is generated by symbols $v\otimes w$ subject to bilinearity. It is characterized by the property that every bilinear map from $V\times W$ factors uniquely through a linear map from $V\otimes W$.

A uniform dilation about the origin multiplies every vector by one positive real scale $s$. The linear map $\mathbf{x}\mapsto s\mathbf{x}$ multiplies lengths by $s$ and $d$-dimensional volumes by $s^d$. This differs from stretching different axes by different factors. The case $s=1$ is the identity map.

For vector spaces $V_1,\dots ,V_r$, every multilinear map $F:V_1\times \cdots \times V_r\to W$ factors uniquely through a linear map $\tilde{F}:V_1\otimes \cdots \otimes V_r\to W$ sending a decomposable tensor to $F$ of its factors. Thus multilinear formulas define linear maps on tensor products without choosing a basis. Tensor contraction is an example, using the evaluation pairing between a vector and a covector.

The sign class of a $d$-dimensional real vector space of functions has VC dimension at most $d$. If $n>d$ points were shattered, the evaluation map from the function space to ${\mathbb{R}}^n$ would have a nonzero vector orthogonal to its image. Choosing labels according to the signs of that vector contradicts orthogonality.