Test data: Latex - statements 1

III.1 Basis

1.1 Definition A basis for a vector space is a sequence of vectors that is linearly independent and that spans the space.

1.5 Definition For any Rn\mathbb{R}^n

En=⟨[10⋮00],[01⋮00],…,[00⋮01]⟩\mathcal{E}_n = \left\langle
\begin{bmatrix}
1 \\ 0 \\ \vdots \\ 0 \\ 0
\end{bmatrix}, \begin{bmatrix}
0 \\ 1 \\ \vdots \\ 0 \\ 0
\end{bmatrix}, \ldots, \begin{bmatrix}
0 \\ 0 \\ \vdots \\ 0 \\ 1
\end{bmatrix} \right\rangle

is the standard (or natural) basis. We denote these vectors e⃗1,e⃗2,…,e⃗n\vec{e}_1, \vec{e}_2, \dots, \vec{e}_n.

1.12 Theorem In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements of the subset in one and only one way.

1.13 Definition In a vector space with basis B the representation of ν⃗\vec{\nu} with respect to BB is the column vector of the coefficients used to express ν⃗\vec{\nu} as a linear combination of the basis vectors:

RepB(ν⃗)=[c1c2⋮cn]B\text{Rep}_B(\vec{\nu}) =
\begin{bmatrix}
c_1 \\ c_2 \\ \vdots \\ c_n
\end{bmatrix}_B

where B=⟨β⃗1,…,β⃗n⟩B = \langle \vec{\beta}_1, \dots, \vec{\beta}_n \rangle and ν⃗=c1β⃗1+c2β⃗2+⋯+cnβ⃗n\vec{\nu} = c_1\vec{\beta}_1 + c_2\vec{\beta}_2 + \cdots + c_n\vec{\beta}_n. The cc's are the coordinates of ν⃗\vec{\nu} with respect to BB.

Reference(s)

  • J. Hefferon, Linear Algebra, 4th ed. (Self-published, 2020).