Linear algebra

Check vector-space conditions and connect subspaces, bases, dimension, and linear transformations.

Meaning

Linear algebra studies vector spaces with laws for addition and scalar multiplication, and linear transformations that preserve this structure. Vectors can be polynomials or functions as well as arrows. Bases and dimension describe independent directions needed for representation.

Example

In the plane, and uniquely represent every vector, as in This is a representative example of a two-dimensional space with two basis vectors.Since a(1,0)+b(0,1)=(a,b), reaching (3,−2) requires a=3 and b=−2. Every point is reachable and the coefficients are unique, which is why these two vectors form a basis.A collection of objects is not automatically a vector space. The scalars, addition, and scalar multiplication must be specified and satisfy the required laws.

Practice for this concept

Scope and references

Topics

Vector spaces, bases, dimension, linear transformations, eigenvalues and eigenvectors

Learning goal

Check vector-space conditions and connect subspaces, bases, dimension, and linear transformations.

Helpful background

Explore the vector components, systems of equations, functions, and basic logic needed for the chosen topic.

Subspaces and eigenvectors

These choice exercises do not establish proofs of the general vector-space axioms or mastery of linear algebra.

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