Question

Matrices A= (1st row)[1 2]
(2nd row) [1 -2]
b= (1st row) [4]
(2nd row) [0]..... Write b as a linear combination of column vectors a1 and a2.

Answer 1

\[A= \left[\begin{matrix}1 & 2 \\ 1 & -2\end{matrix}\right] , B= \left(\begin{matrix}4 \\ 0\end{matrix}\right)\]

Answer 2

b=2 a1 + 1 a2 <--- I don't see how that is correct.

Answer 3

^that's the answer but I don't see it. Is a1= \[\left(\begin{matrix}1 \\ 1\end{matrix}\right)\]

Answer 4

So wouldn't it be \[\left(\begin{matrix}1 \\ 1\end{matrix}\right) +\left(\begin{matrix}-2 \\ 2\end{matrix}\right)=b\]

Answer 5

I think I am confused by your notation, do you mean that\[\vec a_1=\binom11\]and\[\vec a_2=\binom2{-2}\]?

Answer 6

yes,column vectors. Thats how I would answer the problem but the "book" provides b=2 a1+ a2...

Answer 7

yes that answer is right...

Answer 8

hah i see.. So all i had to do was find the scalars that would make c1a1+c2a2=b

Answer 9

forming \(\vec b\) as a linear combination of \(\vec a_1\) and \(\vec a_2\) means finding \(A\) and \(B\)\[A\vec a_1+B\vec a_2=\vec b\]

Answer 10

c1=2, c2=1?

Answer 11

I see you have beat me to the punch :)

Answer 12

You the best! talking through it helps. Thanks. Cheers

Answer 13

so it's basically soving a system of linear equations\[A-2B=4\]\[A+2B=0\]
welcome!

Answer 14

I have to ask you something. You're clearly a genius. Do you teach and apply your genius to research/whatever? Or do you mainly educate?

Answer 15

Well thank you very kindly for the compliment, but the fact is I am quite self-taught and do not apply my mathematical skills in many meaningful ways. I am a humble tutor, attempting to enter university. Any ability I have is a result of pure practice :)

Answer 16

Amazing! You will surely do fine entering a University.

Answer 17

Thanks, best of luck to you as well!