4. Vector Equations
2026-01-28
Vectors
A n×1 matrix is called a n dimensional column vector or simply a nD vector:
w=w1w2⋮wn
Sometimes, we will just call as vector if it is clear that is 2D.
Rn is the set of all nD vectors (向量空间) with real number entries (read r-N).
Specifically, we denote 0 as the zero vector (零向量):
0=00⋮0
Operations on Vectors
Vector Addition
Given two vectors u,v∈Rn, their sum u+v is defined as:
u+v=u1+v1u2+v2⋮un+vn
Scalar Multiplication
Given a vector u∈Rn and a scalar c∈R, the scalar multiplication cu is defined as:
cu=cu1cu2⋮cun
Linear Combinations
A linear combination (线性组合) of vectors v1,v2,…,vk∈Rn with scalars c1,c2,…,ck∈R is defined as:
c1v1+c2v2+…+ckvk
To determine whether a vector b∈Rn can be expressed as a linear combination of vectors v1,v2,…,vk∈Rn, we can set up the following equation:
x1v1+x2v2+…+xkvk=b
This equation can be rewritten in augmented matrix form:
v1,1v1,2⋮v1,nv2,1v2,2⋮v2,n……⋱…vk,1vk,2⋮vk,nb1b2⋮bn
For brevity, write this matrix in a way that identifies its columns:
[v1v2…vkb]
Span (张成空间)
If v1,v2,…,vk∈Rn, then the set of all linear combinations of these vectors is denoted by
Span{v1,v2,…,vk}
and is called the subset of Rn spanned by v1,v2,…,vk.
In other words,
Span{v1,v2,…,vk}={c1v1+c2v2+…+ckvk∣c1,c2,…,ck∈R}
b∈Span{v1,v2,…,vk} if and only if the augmented matrix [v1v2…vkb] is consistent.