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Linear combinations and spans

Linear combinations and spans

Linear Combinations: A linear combination involves taking a set of vectors and forming a new vector by multiplying each vector by a coefficient (scalar) and then summing the results. For example, if you have vectors v1,v2,…,vn\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n and scalars a1,a2,…,ana_1, a_2, \ldots, a_n, a linear combination is expressed as:

u=a1v1+a2v2+⋯+anvn\mathbf{u} = a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_n\mathbf{v}_n

Spans: The span of a set of vectors is the collection of all possible linear combinations of those vectors. If you have a set of vectors {v1,v2,…,vn}\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}, the span, denoted as span(v1,v2,…,vn)\text{span}(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n), represents all vectors that can be formed as linear combinations of this set. The span gives insight into the dimensionality and coverage of the vector space formed by the given vectors.

In essence, linear combinations are the building blocks, while spans indicate the space created by those building blocks.

In this article
Part 1: Linear combinations and span

Part 1: Linear combinations and span

Understanding linear combinations and spans of vectors

When studying "Linear combinations and span," focus on the following key points:

  1. Definitions:

    • Linear Combination: A linear combination of vectors is a sum of scalar multiples of those vectors. For vectors v1,v2,…,vnv_1, v_2, \ldots, v_n, a linear combination can be expressed as c1v1+c2v2+…+cnvnc_1v_1 + c_2v_2 + \ldots + c_nv_n, where c1,c2,…,cnc_1, c_2, \ldots, c_n are scalars.

  2. Span:

    • The span of a set of vectors is the set of all possible linear combinations of those vectors. It represents all the points that can be reached by combining the vectors in the set through scalar multiplication and addition.

  3. Properties of Span:

    • The span of a set of vectors can be thought of as a subspace in a vector space.
    • Adding more vectors to a set that already spans a space does not decrease the span; it can only maintain or expand it.
    • The span of the zero vector is just the zero vector itself.

  4. Geometric Interpretation:

    • In a geometric context, the span can represent a line (1D), a plane (2D), or a higher-dimensional space depending on the number of vectors and their linear independence.

  5. Linear Independence:

    • A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others. If a set of vectors is linearly independent, then their span is maximal in the sense that it cannot be expanded by adding more vectors without losing independence.

  6. Applications:

    • Understanding linear combinations and span is crucial in solving systems of linear equations, understanding dimensions of vector spaces, and in applications such as computer graphics, machine learning, and more.

  7. Example Problems:

    • Practice identifying whether a vector can be expressed as a linear combination of a given set and determining the span of different sets of vectors.

By mastering these key points, you will have a solid understanding of linear combinations and span in linear algebra.

Related topics

Vectors

Linear dependence and independence

Subspaces and the basis for a subspace

Vector dot and cross products

Matrices for solving systems by elimination

Null space and column space

Part 1: Linear combinations and span