Linear Algebra · Lesson 08

Chapter practice

Bring the chapter together: work in a vector space, test a subset, judge independence, recognise a basis, read coordinates, count a dimension and a rank. No worked example sits above the answers.

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span (v_{1}, \dots, v_{n})

dim (V)

v = i \sum c_{i} b_{i}

ker (A)

0 \in V

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Intuition

Bring the chapter together: work in a vector space, test a subset, judge independence, recognise a basis, read coordinates, count a dimension and a rank. No worked example sits above the answers.

Sorting a toolbox means knowing which tool each job needs. These questions differ mainly in which of the chapter's ideas they are asking about.

Before you practise

Decide first which question is being asked. A subspace test looks for the origin and closure; independence asks whether only zero weights give zero; a basis asks for both spanning and independence; a dimension counts a basis; a rank counts the dimension of the column space. The questions are the same in every vector space, and in $\mathbb{R}^m$ most of them are answered by one row reduction.

$c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k=\mathbf{0}$ : the relation independence forbids.

Reminders

A subset missing $\mathbf{0}$ is never a subspace.
A list containing $\mathbf{0}$ , or with one vector a multiple of another, is dependent.
Every basis of a space has the same size, and that size is the dimension.
$A\mathbf{x}=\mathbf{b}$ is solvable exactly when $\mathbf{b}$ lies in the column space.
The pivot columns of $A$ are a basis of $\operatorname{Col}A$ , and $\dim\operatorname{Nul}A=n-\operatorname{rank}A$ .

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