Are These Vectors Linearly Independent?

In summary, the problem involves determining whether the set of vectors {r,u,d,x} is linearly independent or dependent. If the vectors are dependent, a non-trivial linear relation needs to be found. By setting 0=-4*r-4*d-4*u+1*x, it is shown that the vectors are linearly dependent. The problem then becomes finding the scalar in front of the x vector.
  • #1
halo31
51
0

Homework Statement



Let S={r,u,d,x} be a set of vectors.
If x=4r+4u+4d, determine whether or not the four vectors listed above are linearly independent or linearly dependent. If is dependent, find a non-trivial linear relation.

Homework Equations





The Attempt at a Solution



I don't really know how to start the problem. But I tried to set it up this way.
Let 0=4r-4d-4u+x. In order to make it zero would the scalar of x have to be
1? but I am really lost with lost with problem. Any help would great
 
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  • #2
halo31 said:

Homework Statement



Let S={r,u,d,x} be a set of vectors.
If x=4r+4u+4d, determine whether or not the four vectors listed above are linearly independent or linearly dependent. If is dependent, find a non-trivial linear relation.

Homework Equations





The Attempt at a Solution



I don't really know how to start the problem. But I tried to set it up this way.
Let 0=4r-4d-4u+x. In order to make it zero would the scalar of x have to be
1? but I am really lost with lost with problem. Any help would great

You've already written 0 as a linear combination of the vectors {r,u,d,x}. Doesn't that mean you are done??
 
  • #3
well I first have to figure out if its linear dependent or linear independent. I am not given any vectors, just scalars. I am supposed to figure out what scalar number goes in front of the x vector. What I do know when its linear dependent is when there is atleast one nontrivial scalar and when its linear independent is when all scalars are zero. But how do I go about finding what scalar fits in front of the scalar?oh wait wouldn't it be linear dependent since the scalars(4) in front of vectors r,d,u are 4?
 
  • #4
halo31 said:
well I first have to figure out if its linear dependent or linear independent. I am not give any vectors, just scalars. I am supposed to figure out what scalar number goes in front of the x vector.

You are overthinking this. Try putting 1 in front of x.
 
  • #5
That I tried. All I know if I divide it I'm left with the same equation as I started with.
 
  • #6
halo31 said:
That I tried. All I know if I divide it I'm left with the same equation as I started with.

Mmm. I just noticed your algebra is a bit off. If you do it right you have 0=-4*r-4*d-4*u+1*x. All of the scalars in front of the vectors are nonzero and the sum is 0. Linearly dependent or independent??
 
  • #7
ok thanks
 

Related to Are These Vectors Linearly Independent?

1. What is a linear combination?

A linear combination is a mathematical operation where two or more variables are multiplied by a constant and then added together. This is commonly used in linear algebra and is an important concept in fields such as physics and economics.

2. How do I solve linear combination problems?

To solve a linear combination problem, you will need to determine the values of the variables that satisfy the given equations. This can be done through various methods such as substitution or elimination, depending on the complexity of the problem.

3. What is the purpose of linear combination?

Linear combination is used to find relationships between multiple variables. It allows us to find patterns and make predictions based on the given data. It is also a fundamental concept in linear algebra and is essential in solving equations and systems of equations.

4. Can you give an example of a linear combination?

One example of a linear combination is the equation y = 2x + 3, where x and y are variables and 2 and 3 are constants. This equation can be rewritten as y = 2x + 0y + 3, showing how the variables are multiplied by constants and then added together.

5. How is linear combination used in real life?

Linear combination has various applications in real life, such as in economics to determine the relationship between different variables such as supply and demand. It is also used in physics to determine the motion of objects and in statistics to analyze data and make predictions. It is a crucial tool in problem-solving and decision-making processes.

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