Linear Algebra: linear equations

Click For Summary

Homework Help Overview

The discussion revolves around a linear algebra problem involving a linear system represented by Ax = b. Participants are tasked with demonstrating that a specific linear combination of two solutions, u and v, also serves as a solution to the same system.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Some participants suggest substituting the linear combination w into the equation Ax = b to verify its validity. Others express confusion about the substitution process and the relationships between the variables involved.

Discussion Status

The discussion is active, with participants exploring different interpretations of how to approach the problem. Some guidance has been offered regarding the properties of linear transformations and the implications of known solutions.

Contextual Notes

There appears to be some uncertainty regarding the correct application of linearity in the context of the problem, as well as the roles of the variables u, v, and w in the equation.

Amy-Lee
Messages
27
Reaction score
0
Show that if u and v are solutions of the linear system Ax = b, then w = 1/4 u + 3/4 v is also a solution of Ax = b.
 
Physics news on Phys.org
Just plug it in and use the fact that matrix multiplication is linear.
 
Plug what? where?

Ax=b, then b=uv ? and uv=w? uv = 1/4u + 3/4v?
 
You want to show that w "is also a solution of Ax= b".

So "plug" w in for x: Aw= what? Use the fact that w= (1/4)u+ (3/4)v and that A is a linear transformation.
 
Amy-Lee said:
Show that if u and v are solutions of the linear system Ax = b, then w = 1/4 u + 3/4 v is also a solution of Ax = b.
Amy-Lee said:
Plug what? where?

Ax=b, then b=uv ? and uv=w? uv = 1/4u + 3/4v?

Amy-Lee, forget x …

you know that Au = b and Av = b,

so just carry on from there. :wink:
 
tiny-tim said:
Amy-Lee, forget x …

you know that Au = b and Av = b,

so just carry on from there. :wink:

thanks TT, I will try that
 

Similar threads

Null space of dual map of T = annihilator of range T
  • · Replies 1 ·
Replies
1
Views
3K
Orthogonal Projection Problems?
  • · Replies 5 ·
Replies
5
Views
2K
Given specific v, dimension of subspace of L(V, W) where Tv=0?
  • · Replies 15 ·
Replies
15
Views
3K
Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.
  • · Replies 1 ·
Replies
1
Views
2K
Prove that range ##T'## = ##(\text{null}\ T)^0##
  • · Replies 1 ·
Replies
1
Views
1K
Introduction to Linear Algebra: Solving Real-World Problems
  • · Replies 10 ·
Replies
10
Views
2K
Linearly independent vs dependent functions
  • · Replies 3 ·
Replies
3
Views
1K
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
  • · Replies 8 ·
Replies
8
Views
2K
Linear Algebra - Linear (in)dependence of a set
  • · Replies 2 ·
Replies
2
Views
2K
One set v is a linear combination of u. Prove u is linearly dependent
  • · Replies 12 ·
Replies
12
Views
2K