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beketso

  • 3 years ago

linear combinations on vectors.help

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  1. beketso
    • 3 years ago
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    write the matrix\[E= \left[\begin{matrix}3 & 1 \\ 1 & -1\end{matrix}\right]\]

  2. beketso
    • 3 years ago
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    as a linear combination of the matrices \[A=\left[\begin{matrix}1 & 1 \\ 1& 0\end{matrix}\right]\]

  3. beketso
    • 3 years ago
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    \[B=\left[\begin{matrix}0 & 0 \\ 1 & 1\end{matrix}\right]\]

  4. beketso
    • 3 years ago
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    \[C=\left[\begin{matrix}0 & 2 \\ 0 & -1\end{matrix}\right]\]

  5. richyw
    • 3 years ago
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    are there just three?

  6. beketso
    • 3 years ago
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    basically ,you write the matrix E as a linear combination of the matrices A,B, and C

  7. beketso
    • 3 years ago
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    @richyw ,yes there only three

  8. richyw
    • 3 years ago
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    ok so the concepts you need here are matrix addition and scalar multiplication.

  9. richyw
    • 3 years ago
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    you can solve this one in different ways but it's simple enough to do by inspection

  10. richyw
    • 3 years ago
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    so looking at E, the first thing I notice is that \(e_{11}=3\) now looking at matrices A, B, and C. The only one that has anything but 0 in that position is matrix A, so we know already that it must be 3 times matrix A. which gives. \[\left[\begin{matrix}3 & 3 \\ 3 & 0\end{matrix}\right]\] right?

  11. beketso
    • 3 years ago
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    ok i got that part

  12. richyw
    • 3 years ago
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    alright so now look at matrix E and notice that position \(e_{12}\) is 2, so we need to subtract a certain amount from matrix A to produce a 2 there.

  13. richyw
    • 3 years ago
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    well, matrix B has a zero in that position, so it is no good, so subtract one times matrix C from matrix A and we get \[3A-C=\left[\begin{matrix}3&1\\3&1\end{matrix}\right]\]

  14. richyw
    • 3 years ago
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    sorry I should have said "subtract one times matrix C from three times matrix A". Now I hope you can see what multiple of matrix B you bust subtract to get E

  15. richyw
    • 3 years ago
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    are you following?

  16. beketso
    • 3 years ago
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    yeah

  17. richyw
    • 3 years ago
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    alright cool well then you have the answer!

  18. beketso
    • 3 years ago
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    so it is 3A-2B-C=E?

  19. richyw
    • 3 years ago
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    yes!

  20. beketso
    • 3 years ago
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    thanx a lot!!!!!!!

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