6.1 Linear Combination of Vectors

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 6.1.1 Creating Linear Combinations, 6.1.2 Steph Curry Model
Constructions: 6.3 Span of a set of vectors
Generalizations: N/A

Properties: 6.2 Linearity
Sufficiencies: N/A
Questions: N/A

Linear Combination of Vectors

Let $m, n \in N$ . Let a set of 3.1 Column Vector ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{n}$ be

\begin{aligned} {\vec{a}}_{1} = {[\begin{array}{c} a_{1_{1}} \\ a_{2_{1}} \\ ⋮ \\ a_{m_{1}} \end{array}]}_{m \times 1}, & {\vec{a}}_{2} = {[\begin{array}{c} a_{1_{2}} \\ a_{2_{2}} \\ ⋮ \\ a_{m_{2}} \end{array}]}_{m \times 1}, & {\vec{a}}_{n} = {[\begin{array}{c} a_{n_{1}} \\ a_{n_{2}} \\ ⋮ \\ a_{n_{m}} \end{array}]}_{m \times 1} \end{aligned}

Suppose a set of scalars $x_{1}, x_{2}, \dots, x_{n} \in R$ . Then, a linear combination of the vectors ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n}$ is a vector $\vec{b} \in R^{m}$ such that

\vec{b} = x_{1} \cdot {\vec{a}}_{1} + x_{2} \cdot {\vec{a}}_{2} + \dots + x_{n} \cdot {\vec{a}}_{n} = \sum_{k = 1}^{n} x_{k} {\vec{a}}_{k} .

Remark. At it's heart, a linear combination is a vector.
$⧫$

Review

What is a linear combination?
Does a linear combination use scalar or vector addition?
How are linear combinations related to 10 Matrix-Vector Multiplication?

Answers

A linear combination is a sum of scalars being multiplied with vectors.
Vector addition
They're the same thing.

Remark. At it's heart, a linear combination is a vector. ⧫

Remark. At it's heart, a linear combination is a vector.
$⧫$