Solving a Linear Combination Problem

In summary, the conversation involves a student seeking help with a Linear Algebra problem, specifically determining whether a given vector can be written as a linear combination of two other vectors. The student is unsure of advanced methods and has solved the problem using a standard method. They also express difficulty in getting the matrix into reduced row echelon form. A video is suggested as a possible solution.
  • #1
guyvsdcsniper
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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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  • #2
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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  • #3
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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What is a linear combination problem?

A linear combination problem involves finding a solution to a system of linear equations by combining them in a specific way. This means that the variables in the equations must be multiplied by constants and then added together to find a solution that satisfies all of the equations.

What are the steps to solving a linear combination problem?

The steps to solving a linear combination problem include:
1. Identify the variables and constants in the equations
2. Choose one variable to eliminate by multiplying one or both equations by a constant
3. Add or subtract the equations to eliminate the chosen variable
4. Solve for the remaining variable
5. Substitute the solution into one of the original equations to find the value of the eliminated variable
6. Check the solution by plugging it into both equations to make sure it satisfies both of them.

What is the difference between a consistent and an inconsistent linear combination problem?

A consistent linear combination problem has at least one solution that satisfies all of the equations in the system. An inconsistent linear combination problem has no solution that satisfies all of the equations, meaning that the equations are contradictory and cannot be solved simultaneously.

Can a linear combination problem have more than one solution?

Yes, a linear combination problem can have infinitely many solutions. This occurs when the equations in the system are dependent, meaning that they are essentially the same equation. In this case, any value that satisfies one equation will also satisfy the other, resulting in infinitely many solutions.

How can linear combination problems be applied in real life?

Linear combination problems can be used to solve various real-life problems, such as finding the optimal combination of ingredients in a recipe, determining the most efficient way to mix different chemicals, or calculating the best combination of investments to maximize profits. They can also be used in economics to analyze supply and demand or in engineering to optimize designs.

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