Solving a Linear Combination Problem

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SUMMARY

The discussion centers on solving a Linear Algebra problem involving determining if a vector b can be expressed as a linear combination of vectors a1 and a2. The user is struggling to convert a matrix into reduced row echelon form (rref) and seeks confirmation on their method. Participants agree that the user's approach is standard for this type of problem and suggest using row operations to achieve rref. A video resource is recommended for further clarification on the method.

PREREQUISITES
  • Understanding of Linear Algebra concepts, specifically linear combinations.
  • Familiarity with matrix operations, including row reduction.
  • Knowledge of reduced row echelon form (rref) and its significance.
  • Basic proficiency in using mathematical software or tools for matrix manipulation.
NEXT STEPS
  • Watch the recommended video on row operations for achieving rref.
  • Practice solving linear combination problems using different methods.
  • Explore software tools like MATLAB or Python's NumPy for matrix operations.
  • Study the properties and applications of linear combinations in various mathematical contexts.
USEFUL FOR

Students new to Linear Algebra, educators teaching matrix operations, and anyone seeking to improve their problem-solving skills in linear combinations and matrix manipulation.

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Homework Statement
Determine whether b can be written as a linear combination of and . In other words, determine whether weights x1 and x2 exist, such that . Determine the weights and if possible
Relevant Equations
x1a1+x2a2=b
I have attached my work to this thread.

Could someone help me with this Linear Algebra problem. This is my first week so I do not know many advanced ways to solve these problems.

I could not figure out how to get this matrix into rref, so I solved it the following way. Is the way I used appropriate? Is it possible to get this in rref? I am breaking my head trying to think of how I can get it to that form.
 

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I'm not sure what other method you are trying to use for a solution. What you did is a very standard way of solving that problem. I agree with your work.
 
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quittingthecult said:
Homework Statement:: Determine whether b can be written as a linear combination of and . In other words, determine whether weights x1 and x2 exist, such that . Determine the weights and if possible
Relevant Equations:: x1a1+x2a2=b

I have attached my work to this thread.

Could someone help me with this Linear Algebra problem. This is my first week so I do not know many advanced ways to solve these problems.

I could not figure out how to get this matrix into rref, so I solved it the following way. Is the way I used appropriate? Is it possible to get this in rref? I am breaking my head trying to think of how I can get it to that form.
If you want a method in terms of row operations, mayabe this video will help (about 10 mins):
 
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