## beketso 3 years ago linear combinations on vectors.help

1. beketso

write the matrix$E= \left[\begin{matrix}3 & 1 \\ 1 & -1\end{matrix}\right]$

2. beketso

as a linear combination of the matrices $A=\left[\begin{matrix}1 & 1 \\ 1& 0\end{matrix}\right]$

3. beketso

$B=\left[\begin{matrix}0 & 0 \\ 1 & 1\end{matrix}\right]$

4. beketso

$C=\left[\begin{matrix}0 & 2 \\ 0 & -1\end{matrix}\right]$

5. richyw

are there just three?

6. beketso

basically ,you write the matrix E as a linear combination of the matrices A,B, and C

7. beketso

@richyw ,yes there only three

8. richyw

ok so the concepts you need here are matrix addition and scalar multiplication.

9. richyw

you can solve this one in different ways but it's simple enough to do by inspection

10. richyw

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

ok i got that part

12. richyw

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

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

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

are you following?

16. beketso

yeah

17. richyw

alright cool well then you have the answer!

18. beketso

so it is 3A-2B-C=E?

19. richyw

yes!

20. beketso

thanx a lot!!!!!!!