Linear combination of vectors

In summary: Thanks for your co-operation friends.In summary, the conversation revolved around a question about expressing a vector as a linear combination of given vectors. The question was whether it was possible to find values for c1, c2, c3, and c4 that would satisfy the equation v=c1v1+c2v2+c3v3+c4v4, given the vectors v1, v2, v3, and v4. It was determined that there was a problem with the question itself, as the target vector was not in the span of the given vectors, and therefore could not be expressed as a linear combination of them. The conversation also discussed the importance of showing one's work and posting homework questions in the
  • #1
6
0
HI everyone,

v1=[1 4 2 8]^t
v2=[2 5 3 9]^t
v3=[11 14 12 18]^t
v4=[4 3 2 1]^t

I have to express vector v=[7 9 6 8]^t in two ways as a linear combination v=c1v1+c2v2+c3v3+c4v4 of {v1,v2,v3,v4}

Please reply as soon as possible.

Thank You in advance.
 
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  • #2
krocks said:
HI everyone,

v1=[1 4 2 8]^t
v2=[2 5 3 9]^t
v3=[11 14 12 18]^t
v4=[4 3 2 1]^t

I have to express vector v=[7 9 6 8]^t in two ways as a linear combination v=c1v1+c2v2+c3v3+c4v4 of {v1,v2,v3,v4}

Please reply as soon as possible.

Thank You in advance.
Reply how? Tell you the answer? I won't do that! Have you tried anything yourself?

Have you for example, replace v, v1, v,2, v3, and v4 in the equation
v=c1v1+c2v2+c3v3+c4v4 by [7 9 6 8]^t, [1 4 2 8]^t, etc. to get
[7 9 6 8]^t= c1[1 4 2 8]^t+ c2[2 5 3 9]^t+ c3[11 14 12 18]^t+ c4[4 3 2 1]^t.

Go ahead and do the vector calculation on the right and set each component on the left equal to the corresponding component on the right. That will give you four equations to solve for the four numbers c1, c2, c3, and c4.
 
  • #3
Hi HallsOfIvy

Ya i did tried it by myself.
Bu am not able to find values for "c1,c2,c3,c4".

According to me it gives no solution.

So how can I express "v" as linear combination of "v1,v2,v3,v4"?
 
  • #4
As long as you refuse to show what you have done, we cannot see or explain what you have done wrong! I see no problem with solving the equations
c1+ 2c2+ 11c3+ 5c4= 7
4c1+ 5c2+ 14c3+ 3c4= 9
2c1+ 3c2+ 12c3+ 2c4= 6
8c1+ 9c2+ 18c3+ c4= 8
 
  • #5
HallsofIvy said:
As long as you refuse to show what you have done, we cannot see or explain what you have done wrong! I see no problem with solving the equations
c1+ 2c2+ 11c3+ 5c4= 7
4c1+ 5c2+ 14c3+ 3c4= 9
2c1+ 3c2+ 12c3+ 2c4= 6
8c1+ 9c2+ 18c3+ c4= 8
I do. Two problems. That 5 in the first row should be 4, and more importantly, that you had a typo in that particular element doesn't matter.

krocks, are you sure you have the numbers correct?
 
  • #6
Hi

Ya the question i mentioned is absolutely correct.
The equation is:

7=c1+2c2+11c3+4c4
9=4c1+ 5c2+ 14c3+ 3c4
6=2c1+3c2+12c3+2c4
8=8c1+9c2+18c3+c4

This equation is showing no answer becuse the matrix formed by using coefficients of "c1,c2,c3,c4" is singular.

So is there any way to represent "v" as linear combination of "v=c1v1+c2v2+c3v3+c4v4" ??
 
  • #7
The matrix is indeed singular. What is its null space?
 
  • #8
Hey
Please reply friends.

Just tell me the answer only. i am in need of it.
Please
 
  • #9
Looks to me like there is an error in the problem.

After correcting my previous mis-copy, I get the augmented matrix
[tex]\begin{bmatrix}1 & 2 & 11 & 4 & 7 \\ 4 & 5 & 14 & 3 & 9 \\ 2 & 3 & 12 & 2 & 6 \\ 8 & 9 & 18 & 1 & 8\end{bmatrix}[/tex]

But after row-reducing, I get
[tex]\begin{bmatrix}1 & 2 & 11 & 4 & 7 \\ 0 & 1 & 10 & \frac{13}{3} & \frac{19}{3} \\ 0 & 0 & 0& 1& 1\\ 0 & 0 & 0 & 0 & 1\end{bmatrix}[/tex]
and, because of that "1" in the last row, there is no solution. Are you sure you have all of the numbers right? Those vectors are not independent so do not span all of R4 and the "target" vector, <7, 9, 6, 8>, is not in their span- it cannot be written as a linear combination of the given vectors.

If it were in their span, then, because they are not independent, you would be able to write it as a linear combination of them in infinitely many ways.
 
  • #10
Hey HALLSOFIVY

Thanks a ton for help. The question which i got from my professor is exactly the same which I mentioned. I think there's some problem in question itself. i will ask about it from my professor and will post the reply soon here.

Thanks :)
 
  • #11
krocks said:
Hey HALLSOFIVY

Thanks a ton for help. The question which i got from my professor is exactly the same which I mentioned. I think there's some problem in question itself. i will ask about it from my professor and will post the reply soon here.

Thanks :)

This was schoolwork? Why wasn't it posted under the homework section?

You should talk to Derillo- he posted exactly this question under the homework section!
 
  • #12
This row operation is not difficult. Even you got the answer, you better try it again by yourself.
 
  • #13
Hi

I apologize for posting a homework question here.
Actually i was new to this site that's why i wasn't aware of such rules.

But i'll keep that in mind in future
 

What is a linear combination of vectors?

A linear combination of vectors is an expression that combines two or more vectors by multiplying each vector by a scalar (a real number) and then adding the results together. The scalar coefficients determine the magnitude and direction of each vector in the combination.

What is the purpose of finding a linear combination of vectors?

The purpose of finding a linear combination of vectors is to determine if a given vector can be written as a combination of other vectors. This allows for a deeper understanding of the relationships between different vectors and can be useful in solving systems of linear equations.

How do you determine if a vector is a linear combination of other vectors?

To determine if a vector is a linear combination of other vectors, you can set up a system of equations where the coefficients of the vectors in the combination are the variables. If the system has a unique solution, then the vector is a linear combination of the other vectors. If the system has no solution, then the vector is not a linear combination of the other vectors.

Can a linear combination of vectors result in a zero vector?

Yes, a linear combination of vectors can result in a zero vector if all of the vectors in the combination cancel each other out. This can happen when the scalar coefficients are carefully chosen to create a balance between the vectors.

How is a linear combination of vectors used in real-world applications?

Linear combinations of vectors are used in various fields of science and engineering, such as physics, computer graphics, and machine learning. They can be used to model complex systems, analyze data, and make predictions. For example, in physics, linear combinations of force vectors can be used to determine the net force acting on an object. In computer graphics, linear combinations of color vectors can be used to create different shades and hues. In machine learning, linear combinations of features can be used to classify data points.

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