本次日本作业案例分享主要为线性代数代写的assignment

1. 矩阵的核
定义 1.1。一个线性方程组称为齐次方程组，如果
每个方程中的常数项都为零，即如果系统可以写成
作为：
！
” ” #
” ” $
a1,1×1 + a1,2×2 +… + a1,nxn =0
a2,1×1 + a2,2×2 +… + a2,nxn =0
………………
am,1×1 + am,2×2 +… + am,nxn =0
齐次系统对应于矩阵方程：
一个！” x = ！” 0
在哪里
!” 0 是 Rm 的零向量，A 是系统的系数矩阵。
增广矩阵 B 的最后一列为零。这样的系统总是一致的，
因为零向量，对应于
！” x = ！” 0
（即 x1 = x2 = … = xn = 0）总是一个解。
提案 1.2。一个齐次的线性方程组要么只有一个
解，Rn 的零向量，称为平凡解，或有
无穷多个解。
定义 1.3。设 A 是一个 m#n 矩阵。的核或零空间
矩阵 A 是齐次系统 A 的解空间！” x = 0. 记为
由 Ker(A)。
例 1.4。放
一 =
%
21
11
&
A的核是系统的解集：
‘
2×1 + x2 =0
x1 + x2 =0
我们求解这个系统，我们很容易得到：
x1 = x2 =0。
换句话说，
!” x 是零向量。所以这里，A 的核只包含
零向量：
克尔（A）= {
！”0}。
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1. The kernel of a matrix
Definition 1.1. A system of linear equations is called homogeneous if the con-
stant term in each equation is equal to zero, that is, if the system can be written
as:
!
” ” #
” ” $
a1,1×1 + a1,2×2 +… + a1,nxn =0
a2,1×1 + a2,2×2 +… + a2,nxn =0
… … … … …
am,1×1 + am,2×2 +… + am,nxn =0
A homogeneous system correponds to the matrix equation:
A!” x = !” 0
where
!” 0 is the zero vector of Rm and A is the coe!cient matrix of the system. The
last column of the augmented matrix B is zero. Such a system is always consistent,
because the zero vector, corresponding to
!” x = !” 0
(that is x1 = x2 = … = xn = 0) is always a solution.
Proposition 1.2. A homogeneous system of linear equations either has only one
solution, the zero vector of Rn, which is called the trivial solution, or it has
infinitely many solutions.
Definition 1.3. Let A be a m # n matrix. The kernel or the nullspace of the
matrix A is the space of solutions of the homogeneous system A!” x = 0. It is denoted
by Ker(A).
Example 1.4. Set
A =
%
21
11
&
The kernel of A is the set of solutions of the system:
‘
2×1 + x2 =0
x1 + x2 =0
We solve this system and we easily obtain:
x1 = x2 =0.
In other words,
!” x is the zero vector. So here, the kernel of A contains only the
zero vector:
Ker(A)= {
!” 0 }.
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