- #1

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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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- Thread starter krocks
- Start date

- #1

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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.

- #2

HallsofIvy

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Reply how? Tell you the answer? I won't do that! Have you tried anything yourself?

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.

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

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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

HallsofIvy

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c1+ 2c2+ 11c3+ 5c4= 7

4c1+ 5c2+ 14c3+ 3c4= 9

2c1+ 3c2+ 12c3+ 2c4= 6

8c1+ 9c2+ 18c3+ c4= 8

- #5

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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.

c1+ 2c2+ 11c3+ 5c4= 7

4c1+ 5c2+ 14c3+ 3c4= 9

2c1+ 3c2+ 12c3+ 2c4= 6

8c1+ 9c2+ 18c3+ c4= 8

krocks, are you sure you have the numbers correct?

- #6

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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

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The matrix is indeed singular. What is its null space?

- #8

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Hey

Please reply friends.

Just tell me the answer only. i am in need of it.

Please

Please reply friends.

Just tell me the answer only. i am in need of it.

Please

- #9

HallsofIvy

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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 R

If it were in their span,

- #10

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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

HallsofIvy

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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

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This row operation is not difficult. Even you got the answer, you better try it again by yourself.

- #13

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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

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