Linear independence

Key takeaways

Linear independence belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear independence to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear independence from memory before moving on to harder problems.

Reference excerpt

In linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the other vectors in the set. If such a vector exists, then the vectors are said to be linearly dependent. Linear independence is part of the definition of linear basis. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Definition A sequence of vectors v 1 , v 2 , … , v k {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{k}} from a vector space V is said to be linearly dependent, if there exist scalars a 1 , a 2 , … , a k , {\displaystyle a_{1},a_{2},\dots ,a_{k},} not all zero, such that

a 1 v 1 + a 2 v 2 + ⋯ + a k v k = 0 , {\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} ,}

where 0 {\displaystyle \mathbf {0} } denotes the zero vector. If ⁠ k = 1 {\displaystyle k=1} ⁠, this implies that a single vector is linear dependent if and only if it is the zero vector. If ⁠ k > 1 {\displaystyle k>1} ⁠, this implies that at least one of the scalars is nonzero, say a 1 ≠ 0 {\displaystyle a_{1}\neq 0} , and the above equation is able to be written as

v 1 = − a 2 a 1 v 2 + ⋯ + − a k a 1 v k . {\displaystyle \mathbf {v} _{1}={\frac {-a_{2}}{a_{1}}}\mathbf {v} _{2}+\cdots +{\frac {-a_{k}}{a_{1}}}\mathbf {v} _{k}.}

Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others. A sequence of vectors v 1 , v 2 , … , v n {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}} is said to be linearly independent if it is not linearly dependent, that is, if the equation

a 1 v 1 + a 2 v 2 + ⋯ + a n v n = 0 , {\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{n}\mathbf {v} _{n}=\mathbf {0} ,}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Linear independence

Start with the simplest possible case. Write down what Linear independence claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear independence before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear independence ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear independence

In research: Linear independence appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear independence in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear independence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstract algebra, Linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Linear independence outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Linear independence in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear independence means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear independence out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear independence in simple terms?

In linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the other vectors in the set. If such a vector exists, then the vectors are said to be linearly dependent.

Why does Linear independence matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear independence?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear independence.