Subspace And All Solutions to Ax=b

本文深入探讨了线性代数中子空间的概念,包括子空间定义、列空间、行空间、零空间以及秩与简化阶梯形矩阵的关系。通过具体实例解析,详细介绍了如何求解Ax=0和Ax=b方程组,揭示了子空间维度、自由变量与系统解之间的联系。

摘要生成于 C知道 ,由 DeepSeek-R1 满血版支持, 前往体验 >

Subspace


The Definition Of Subspace

  • A subspace of a vector space is a set of vectors (including 0) that satisfied two requirments: if www and vvv are verctor in the subspace and ccc is any sclar,then
  1. v+wv+wv+w is in the subspace
  2. cvcvcv in the subspace

  • Every subspace must contain the zero vector. for example:
  1. The plane in R3R^3R3 has to go through ( 0, 0, 0)
  2. Lines through the origin are also subspace.



The Column Space of AAA

  • The Column space consists of all linear combinations of the columns. The combinations are all possible vectors AxAxAx. They fill the column space C(A)C(A)C(A)

  • The system Ax=bAx=bAx=b is solvable if and only if b is in the column space of AAA

  • Suppose AAA is a m by n matrix , columns belongs to RmR^mRm (每个column 是m个元素), the column space of AAA is a subspace of RmR^mRm



The Row Space of AAA

  • Like column space , row space is a linear conbinations of rows,rows belongs to RnR^nRn(每一行有n个元素),所以row space 是 RnR^nRn的subspace



The Null Space of AAA

  • There are nonzero solutions to Ax=0Ax=0Ax=0 , each solution x belongs to the nullspace of A, which is denoted by N(AAA) .

  • The solution vector x have n components. they are vectors in RnR^nRn , so nullspace is a subspace of RnR^nRn

  • Solving Ax=0Ax=0Ax=0 by elimination

Ax=0        −>     [112322810331013][x1x2x3x4]=[0000] Ax=0 \space\space\space\space\space\space\space\space->\space\space\space\space\space\left[ \begin{matrix} 1 & 1&2 & 3 \\ 2 & 2 & 8 &10\\ 3 & 3 & 10& 13 \end{matrix} \right] \left[ \begin{matrix} x1 \\ x2 \\ x3 \\ x4 \end{matrix} \right]= \left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right] Ax=0        >     123123281031013x1x2x3x4=0000


Get Triangular U by eliminating A: U=[112300440000]U= \left[ \begin{matrix} 1 & 1&2 & 3 \\ 0 & 0 & 4 &4\\ 0 & 0 & 0& 0 \end{matrix} \right]U=100100240340


The pivot variables are x1x1x1 and x3x3x3 : column1 and 3 contains pivots
The free variables are x2x2x2 and x4x4x4: column2 and 4 have no pivots


Special solutions:

  • set x2=1,x4=0x2=1 ,x4=0x2=1,x4=0 , by back substitution x3=0x3=0x3=0 ,then x1=−1x1=-1x1=1
  • set x2=0,x4=1x2=0 ,x4=1x2=0,x4=1 , by back substitution x3=−1x3=-1x3=1 ,then x1=−1x1=-1x1=1

Complete solutions to Ax=0Ax=0Ax=0:  x2x2x2 and x4x4x4 can be any multiplies
x=x2[−1100]+x4[−10−11]x=x2\left[ \begin{matrix} -1 \\ 1 \\ 0 \\ 0 \end{matrix} \right]+x4\left[ \begin{matrix} -1 \\ 0 \\ -1 \\ 1 \end{matrix} \right]x=x21100+x41011

  • With n>m ,there is at least one free variable . The system Ax=0Ax=0Ax=0 has at least one nonzero solution. the nulllspace dimension is the number of free variables(at least n-m).



The Rank And Reduced Echelon Form R

  • The rank of AAA is the number of pivots ,this number is r

  • The rank r is the “dimension” of the column space. It is also the dimension of the row space . The great thing is that r also reveals the dimetion of nullspace(n-r).

  • R:the conponents above the pivots are 0 in the pivot columns (column 1 and column 3 )
    A=[1302−10014−31316−4]yields R=[1302−10014−300000]A= \left[ \begin{matrix} 1 & 3&0 & 2&-1\\ 0 & 0 & 1 &4&-3\\ 1 & 3 & 1& 6&-4 \end{matrix} \right] yields \space R=\left[ \begin{matrix} 1 & 3&0 & 2&-1\\ 0 & 0 & 1 &4&-3\\ 0 & 0 & 0& 0&0 \end{matrix} \right]A=101303011246134yields R=100300010240130

The form of R is like :[IF00]\left[ \begin{matrix} I &F \\ 0 & 0\\ \end{matrix} \right][I0F0]
Nullspace matrix form is like :[−FI]\left[ \begin{matrix} -F \\ I\\ \end{matrix} \right][FI]


Nullspace for the A(5-2=3) is:[−3−211000−43010001][x1 not freex2 freex3 not freex4 freex5 free]\left[ \begin{matrix} -3 & -2&1 \\ 1 & 0 & 0 \\ 0 & -4 & 3\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]\left[ \begin{matrix} x1 \space not\space free \\ x2 \space free \\ x3 \space not\space free \\ x4 \space free \\ x5 \space free \\ \end{matrix} \right]310002041010301x1 not freex2 freex3 not freex4 freex5 free




All solutions to Ax=bAx=bAx=b


Ax=b

[130200141316][x1x2x3x4]=[167]\left[ \begin{matrix} 1 & 3&0&2\\ 0 & 0 & 1 &4\\ 1 & 3 & 1& 6 \end{matrix} \right]\left[ \begin{matrix} x1 \\ x2\\ x3 \\ x4 \end{matrix} \right]=\left[ \begin{matrix} 1 \\ 6\\ 7 \end{matrix} \right]101303011246x1x2x3x4=167


augmented matrix:
[A b]=[130210014613167][A \space b]=\left[ \begin{matrix} 1 & 3&0&2&1\\ 0 & 0 & 1 &4&6\\ 1 & 3 & 1& 6&7 \end{matrix} \right][A b]=101303011246167


after elimination:
[130210014600000]=[R d]\left[ \begin{matrix} 1 & 3&0&2&1\\ 0 & 0 & 1 &4&6\\ 0 & 0 &0& 0&0 \end{matrix} \right]=[R\space d]100300010240160=[R d]


particular solution:set x2=x4=0 then Xp=(1, 0, 6, 0)
the solutions to Ax=0:Xn=[−3−2100−401]Xn=\left[ \begin{matrix} -3& -2\\ 1 & 0 \\ 0 & -4 \\ 0&1 \end{matrix} \right]Xn=31002041

Complete solutions:X=Xp+Xn
X=[1060]+x2[−3100]+x4[−20−41]X=\left[ \begin{matrix} 1\\ 0 \\ 6 \\ 0 \end{matrix} \right]+x2\left[ \begin{matrix} -3\\ 1 \\ 0 \\ 0 \end{matrix} \right]+x4\left[ \begin{matrix} -2\\ 0 \\ -4 \\ 1 \end{matrix} \right]X=1060+x23100+x42041


four possibilities of solutions depend on rank r


rAsolutions
r=m and r=nsquare and invertibleAx=b has 1 solution
r=m and r<nshort and wideAx=b has ∞\infty solutions
r<m and r=ntall and thinAx=b has 0 or 1 solution
r<m and r<nnot full rankAx=b has 0 or ∞\infty solutions
### 子空间干扰中的目标检测设计与性能分析 近年来,深度学习方法已经在多个领域产生了深远影响,包括自然图像理解、语音识别等[^1]。然而,在特定的技术挑战方面,如子空间干扰下的目标检测,研究工作也取得了显著进展。 #### 设计原理 在存在子空间干扰的情况下,目标检测面临的主要问题是区分真实信号和背景噪声的能力下降。为此,研究人员提出了多种算法来增强系统的鲁棒性和准确性: - **特征提取优化**:通过引入更复杂的卷积神经网络结构,可以有效提高模型对于复杂环境的理解能力。 - **多模态数据融合**:结合来自不同传感器的数据源(例如雷达和摄像头),能够提供更加全面的信息支持,从而改善最终决策的质量。 ```python import numpy as np def feature_extraction(image_data): # 假设这是一个用于处理图像并提取特征的函数 features = model.predict(image_data) return features ``` #### 性能评估指标 为了衡量这些改进措施的效果,通常会采用一系列量化标准来进行测试对比实验。常见的评价维度包括但不限于以下几个方面: - **精度(Precision)** 和 **召回率(Recall)** :这两个参数反映了分类器做出正确判断的比例以及它找到所有正样本的能力。 - **F1分数(F1 Score)** : 综合考虑了Precision和Recall之间的平衡关系,适用于不平衡类别分布的情况。 - **平均绝对误差(Mean Absolute Error, MAE)** 或者 **均方根误差(Root Mean Square Error, RMSE)** : 当预测值为连续变量时适用;前者计算实际值与估计值之间差异的平均大小,后者则强调较大偏差的影响程度。 早期的研究成果已经为进一步探索奠定了基础,并预示着未来可能取得更大突破的方向[^2]。
评论
添加红包

请填写红包祝福语或标题

红包个数最小为10个

红包金额最低5元

当前余额3.43前往充值 >
需支付:10.00
成就一亿技术人!
领取后你会自动成为博主和红包主的粉丝 规则
hope_wisdom
发出的红包
实付
使用余额支付
点击重新获取
扫码支付
钱包余额 0

抵扣说明:

1.余额是钱包充值的虚拟货币,按照1:1的比例进行支付金额的抵扣。
2.余额无法直接购买下载,可以购买VIP、付费专栏及课程。

余额充值