Linear Algebra: linear equations

In summary, the conversation discusses how to 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. It is suggested to plug in w for x and use the fact that w is equal to 1/4 u + 3/4 v and A is a linear transformation. This will result in Aw = b, which proves that w is a solution of Ax = b.
  • #1
Amy-Lee
27
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
  • #2
Just plug it in and use the fact that matrix multiplication is linear.
 
  • #3
Plug what? where?

Ax=b, then b=uv ? and uv=w? uv = 1/4u + 3/4v?
 
  • #4
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.
 
  • #5
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:
 
  • #6
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
 

1. What is a linear equation in terms of linear algebra?

A linear equation is a mathematical expression that represents a straight line on a graph. In linear algebra, it is typically written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

2. How do you solve a system of linear equations?

To solve a system of linear equations, you must first determine the number of variables and equations in the system. Then, you can use techniques such as substitution, elimination, or matrices to find the values of the variables that satisfy all of the equations.

3. What is the difference between a linear equation and a nonlinear equation?

A linear equation has a constant slope and forms a straight line when graphed. A nonlinear equation does not have a constant slope and does not form a straight line when graphed. Nonlinear equations can have curves or other shapes on a graph.

4. Can linear equations have more than two variables?

Yes, linear equations can have any number of variables. However, the number of equations must be equal to or greater than the number of variables in order for the system to be solvable.

5. What is the importance of linear algebra in real-world applications?

Linear algebra is used extensively in fields such as engineering, physics, computer science, and economics to model and solve real-world problems. It provides a powerful toolset for analyzing and manipulating data, making it a crucial component of many scientific and technological advancements.

Similar threads

Null space of dual map of T = annihilator of range T
    • Calculus and Beyond Homework Help
Replies
0
Views
441
Given specific v, dimension of subspace of L(V, W) where Tv=0?
    • Calculus and Beyond Homework Help
Replies
14
Views
582
Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.
    • Calculus and Beyond Homework Help
Replies
1
Views
599
Orthogonal Projection Problems?
    • Calculus and Beyond Homework Help
Replies
5
Views
935
Prove that range ##T'## = ##(\text{null}\ T)^0##
    • Calculus and Beyond Homework Help
Replies
1
Views
449
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
    • Calculus and Beyond Homework Help
Replies
8
Views
607
Linearly independent functions with identically zero Wronskian
    • Calculus and Beyond Homework Help
Replies
1
Views
257
Do these two statements imply an underlying induction proof?
    • Calculus and Beyond Homework Help
Replies
7
Views
401
One set v is a linear combination of u. Prove u is linearly dependent
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Introduction to Linear Algebra: Solving Real-World Problems
    • Calculus and Beyond Homework Help
Replies
10
Views
993

Hot Threads

Recent Insights

Back
Top