Linear algebra

前置知识

20 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

• 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.

通向

可选发现

• 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

学习路径

• Accelerated secondary progression
• Undergraduate mathematics major
• Applied mathematics

延伸阅读

• Sheldon Axler, Linear Algebra Done Right (1996)
• Gilbert Strang, Introduction to Linear Algebra