Early undergraduateAlgebraPlanned
Linear algebra
Systems of linear equations and Gaussian elimination, matrices, vector spaces, linear maps, bases and dimension, rank–nullity, determinants, eigenvalues and diagonalisation.
math.algebra.linear-algebra
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 20
You will learn to
- Solve linear systems by Gaussian elimination and describe solution sets as translates of null spaces.
- Prove that all bases of a finite-dimensional vector space have the same size, and apply the rank–nullity theorem.
- Represent linear maps by matrices, compose them by matrix multiplication and change basis.
- Compute eigenvalues and eigenvectors and decide whether a matrix is diagonalisable.
Definition
- A vector space over a field k is a set with addition and scalar multiplication satisfying the vector-space axioms.
- A linear map T : V → W satisfies T(u + v) = T(u) + T(v) and T(cv) = cT(v).
- A basis is a linearly independent spanning set; dim V is the number of elements in a basis.
- λ is an eigenvalue of T if Tv = λv for some non-zero vector v.
Theorem
- Rank–nullity: dim V = dim ker T + dim im T for a linear map T : V → W with V finite-dimensional.
- A composite of linear maps is linear, and corresponds to the product of their matrices.
- A square matrix is invertible if and only if its determinant is non-zero, if and only if its columns are linearly independent.
- Eigenvectors belonging to distinct eigenvalues are linearly independent.
Common misconceptions
- Thinking matrix multiplication is commutative.
- Assuming every square matrix is diagonalisable; the matrix with rows (0, 1) and (0, 0) is not.
- Believing a linear map must send lines through the origin to lines; it may collapse them to a point.
Leads to
Optional discoveries
- Categories Vector spaces and linear maps form a category: composites of linear maps are linear and identity maps are linear.
- Representation theory of finite groups A representation of a group is a homomorphism into the group of invertible linear maps of a vector space.
- ← Linear equations and inequalities
- ← The coordinate plane
Learning pathways
Further reading
- Sheldon Axler, Linear Algebra Done Right (1996)
- Gilbert Strang, Introduction to Linear Algebra