MIT18.06学习笔记 - Lecture 9: Independence, basis and dimension

这个系列文章是我重温Gilbert老爷子的线性代数在线课程的学习笔记。
Course Name：MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章节内容: 3.5

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课程提纲
1. Linear Independence
2. Vectors span a subspace and basis
3. Dimension of a vector space
4. Bases for Matrix Spaces

课程重点

Linear Independence

Vectors span a subspace and basis

Definition: A set of vectors spans a space if their linear combinations fill the space.
Row space: the row space of a matrix is the subspace of $R^n$ spanned by the rows. The row space of $A$ is $C(A^T)$. It is the column space of $A^T$.

Basis

This combination of properties is fundamental to linear algebra.





The column spaces of $A$ and $R$ are different. Their bases are different, but their dimensions are the same. The row space of $A$ is the same as the row spaces of $R$.

Dimension of a vector space

Definition: The dimension of a space is the number of vectors in every basis.

Note about the language of linear algebra: rank is for matrix, dimension and basis is for space.

Bases for Matrix Spaces

matrix space and three subspaces: $dim(S) + dim(U) = dim(S \cap U) + dim(S + U)$