# Linear combination

A linear combination is an expression built from a set of vectors by multiplying each one by a scalar and adding the results: for vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k \) in a vector space over a field \( F \), any vector of the form

\[ \mathbf{v}_1 c_1 + \mathbf{v}_2 c_2 + \cdots + \mathbf{v}_k c_k, \qquad \text{where each } c_1, \ldots, c_k \text{ lies in } F \]

is a linear combination of them.<sup>[1](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)</sup> The scalars \( c_i \) may be zero, negative, or complex when \( F = \mathbb{C} \). The concept is the constructive core of linear algebra: spans, linear independence, bases, and coordinates are all statements about which linear combinations exist and when they are unique.

| Key fact | Detail |
|---|---|
| Definition | \( c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k \) with scalars \( c_i \) in the field \( F \) (typically \( \mathbb{R} \) or \( \mathbb{C} \))<sup>[1](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)</sup> |
| Span | The set of all linear combinations of \( \mathbf{v}_1, \ldots, \mathbf{v}_k \), denoted \( \operatorname{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} \)<sup>[1](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)</sup> |
| Dependence test | A set is linearly dependent if and only if at least one of its vectors is a linear combination of the others<sup>[2](https://www.math.u-bordeaux.fr/~eprovenzi/include/LinearAlgebra.pdf)</sup> |
| Independence and uniqueness | If \( \mathbf{v}_1, \ldots, \mathbf{v}_n \) are linearly independent, every vector in their span can be expressed as a linear combination of them in at most one way<sup>[3](https://people.tamu.edu/~fulling/linearity/lin4.pdf)</sup> |
| Basis | A basis is a linearly independent set whose linear combinations fill the whole space; Grassmann showed any vector is uniquely expressible through the elements of a basis<sup>[4](https://www.math.mcgill.ca/darmon/courses/history/katz.pdf)</sup> |
| Beyond vectors | The same construction applies to functions, matrices, and any objects forming a vector space, such as \( 3e^{2x} + 4\sin(3x) - 3\cos(3x) \) among continuous functions<sup>[5](https://opentext.uleth.ca/PDF/3410notes-ebook.pdf)</sup> |

## Span

The span of \( \mathbf{v}_1, \ldots, \mathbf{v}_k \) is the collection of all linear combinations of those vectors, written \( \operatorname{Span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} \).<sup>[1](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)</sup><sup> • </sup><sup>[6](https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_%28Margalit_and_Rabinoff%29/02%3A_Systems_of_Linear_Equations-_Geometry/2.02%3A_Vector_Equations_and_Spans)</sup> In the plane, the span of two non-parallel vectors is the whole xy-plane, since every point arises as a linear combination of the two.<sup>[7](https://math.ucdavis.edu/~linear/linear-guest.pdf)</sup>

Since the definition constructs a set through a rule, showing that a vector lies in a span requires displaying the scalars that yield it, while demonstrating that a set sits inside a span calls for finding suitable scalars for any given member of that set.<sup>[1](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)</sup>

Sums of subspaces illustrate how combinations generate new objects: the sum of two distinct lines through the origin in space is strictly bigger than their union, and the vector \( (1, 1, 2) = (1, 0, 1) + (0, 1, 1) \) lies in the sum but on neither line.<sup>[8](https://one-course.com/books/math/3/en/chapter/18-vector-spaces)</sup>

## Linear independence and uniqueness

[Linear independence](https://www.edgepedia.org/linear-independence) is defined through linear combinations: a set \( \{\mathbf{u}_1, \ldots, \mathbf{u}_k\} \) is linearly independent if and only if no nontrivial linear combination of its vectors equals the zero vector, meaning the only solution of \( a_1 \mathbf{u}_1 + \cdots + a_k \mathbf{u}_k = \mathbf{0} \) is \( a_1 = \cdots = a_k = 0 \).<sup>[9](https://dlib.hust.edu.vn/bitstream/HUST/25756/1/OER000005531.pdf)</sup> A set is linearly dependent precisely when at least one vector can be written as a linear combination of the others.<sup>[2](https://www.math.u-bordeaux.fr/~eprovenzi/include/LinearAlgebra.pdf)</sup>

Independence controls uniqueness. If \( \mathbf{v}_1, \ldots, \mathbf{v}_n \) are independent, then every vector in the space can be expressed in at most one way as a linear combination of them.<sup>[3](https://people.tamu.edu/~fulling/linearity/lin4.pdf)</sup>

## Solving for coefficients

Deciding whether a given vector is a linear combination of given vectors amounts to solving a linear system. The vector equation \( x_1 \mathbf{v}_1 + \cdots + x_k \mathbf{v}_k = \mathbf{b} \), with unknowns \( x_1, \ldots, x_k \), asks which \( \mathbf{b} \) lie in the span.<sup>[6](https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_%28Margalit_and_Rabinoff%29/02%3A_Systems_of_Linear_Equations-_Geometry/2.02%3A_Vector_Equations_and_Spans)</sup> A solution exists exactly when \( \mathbf{b} \) is a linear combination of the \( \mathbf{v}_i \), and the solution entries are the coefficients. Typical exercises ask exactly this: determine whether vectors are dependent or independent, express a vector through basis vectors, or find a basis of a spanned space.<sup>[10](https://www.youtube.com/watch?v=MmlaW48WZt0)</sup>

## Beyond column vectors

The definition applies to any vector space, not only lists of numbers. In the space of real-valued continuous functions on \( \mathbb{R} \), an expression such as \( f(x) = 3e^{2x} + 4\sin(3x) - 3\cos(3x) \) is a linear combination of the functions \( e^{2x} \), \( \sin(3x) \), and \( \cos(3x) \).<sup>[5](https://opentext.uleth.ca/PDF/3410notes-ebook.pdf)</sup> Matrices, polynomials, and solutions of differential equations admit the same treatment, with the scalars drawn from the underlying field.<sup>[11](https://www.ee.iitm.ac.in/~andrew/EE5120/VectorSpaceBasics.html)</sup>

## History

The modern vocabulary descends from Hermann Grassmann (1809–1877), whose *Ausdehnungslehre* (Linear Extension Theory, 1844) treated formal linear combinations \( a_1 e_1 + a_2 e_2 + a_3 e_3 + \cdots \) of abstract units \( e_1, e_2, e_3, \ldots \) with real coefficients, and formally proved the linear space properties for these operations.<sup>[12](https://mathshistory.st-andrews.ac.uk//Biographies/Grassmann/)</sup> In the Ausdehnungslehre he introduced the concepts of linear independence and of a basis, demonstrated that every vector has a unique representation as a linear combination built from a basis's elements, and established the exchange theorem whereby, in an \( n \)-dimensional space, a basis vector may be swapped for any other vector independent of the remaining \( n-1 \).<sup>[4](https://www.math.mcgill.ca/darmon/courses/history/katz.pdf)</sup> His extension theory introduced what are now called basis, dimension, generator, and linear dependence and independence, though it found little acceptance in his own time and included no axiomatization.<sup>[13](https://educa.fcc.org.br/scielo.php?lng=en&nrm=iso&pid=S2526-76472024000100302&script=sci_arttext&tlng=en)</sup><sup> • </sup><sup>[14](https://academic.oup.com/book/41041/chapter/349347436)</sup>

The abstract notion of a vector space was first isolated by [Giuseppe Peano](https://www.edgepedia.org/giuseppe-peano) in 1888, in a geometric setting, completing the move from Grassmann's concrete combinations to the axiomatic theory on which modern linear algebra rests.<sup>[15](https://www.sciencedirect.com/science/article/pii/S0315086085710257)</sup>

## References

1. [Linear Combinations and Span, Wichita State University, Math 511 notes](https://courses.math.wichita.edu/math511/ch1/p4/FormalismAndProof.html)
2. [E. Provenzi, Linear Algebra lecture notes, Université de Bordeaux](https://www.math.u-bordeaux.fr/~eprovenzi/include/LinearAlgebra.pdf)
3. [B. Fulling, Linear Algebra notes, Texas A&M University](https://people.tamu.edu/~fulling/linearity/lin4.pdf)
4. [V. J. Katz, A History of Mathematics, McGill University course text](https://www.math.mcgill.ca/darmon/courses/history/katz.pdf)
5. [Linear Algebra notes, University of Lethbridge](https://opentext.uleth.ca/PDF/3410notes-ebook.pdf)
6. [Vector Equations and Spans, Interactive Linear Algebra (Margalit and Rabinoff), LibreTexts](https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_%28Margalit_and_Rabinoff%29/02%3A_Systems_of_Linear_Equations-_Geometry/2.02%3A_Vector_Equations_and_Spans)
7. [Linear Algebra, UC Davis](https://math.ucdavis.edu/~linear/linear-guest.pdf)
8. [Vector Spaces, One Course textbook, chapter 18](https://one-course.com/books/math/3/en/chapter/18-vector-spaces)
9. [Linear Independence as a Linear Combination, HUST open educational resources](https://dlib.hust.edu.vn/bitstream/HUST/25756/1/OER000005531.pdf)
10. [Linear independence and basis vectors, video lecture](https://www.youtube.com/watch?v=MmlaW48WZt0)
11. [Vector Space Basics, EE5120 notes, IIT Madras](https://www.ee.iitm.ac.in/~andrew/EE5120/VectorSpaceBasics.html)
12. [Hermann Grassmann (1809–1877), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Grassmann/)
13. [The Historical Movement of the Contributions of Grassmann's Extension Theory to Linear Algebra, SciELO](https://educa.fcc.org.br/scielo.php?lng=en&nrm=iso&pid=S2526-76472024000100302&script=sci_arttext&tlng=en)
14. [Grassmann's Concept Structuralism, Oxford Academic](https://academic.oup.com/book/41041/chapter/349347436)
15. [The axiomatization of linear algebra: 1875–1940, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0315086085710257)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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