Show that w= [3,-5] can be written as a linear combination of the vectors [2,7] [1,-1] [-2,1]
(All vertical vectors btw)

- anonymous

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- jamiebookeater

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- ganeshie8

like this :
\[
W=
\left[ {\begin{array}{cc}
3 \\
-5\\
\end{array} } \right] ,
A =
\left[ {\begin{array}{cc}
2 \\
7\\
\end{array} } \right],
B =
\left[ {\begin{array}{cc}
1 \\
-1\\
\end{array} } \right],
C =
\left[ {\begin{array}{cc}
-2\\
1\\
\end{array} } \right]
\]

- ganeshie8

you wanto write \( W\) as linearcombination of \(A, B, C\), right ?

- anonymous

yep

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## More answers

- ganeshie8

lets see.. guess we need to find it trial and error

- anonymous

there is way to do it by just putting it int a matrix but the thing is there is 2 equations and 3 unknowns so that is why I am confused

- ganeshie8

W = Ax + By + Cz
you need to find x, y, z

- ganeshie8

so yes, two equations, and 3 unknowns

- ganeshie8

3 = 2x+y-2z --------(1)
-5 = 7x-y+z---------(2)

- ganeshie8

you will get infinite solutions, just pick one ?

- anonymous

oh okay how do you know it will be infinite soln though?

- ganeshie8

look at each of the equation u have,
it has 3 variables eh ?

- anonymous

yep

- ganeshie8

that means, its a plane in 3 dimensions.
2 planes meet in a line

- ganeshie8

a line has infinite points. so infinite solutions

- ganeshie8

lets find them..

- ganeshie8

few of them atleast :)

- anonymous

okay

- anonymous

:)

- ganeshie8

3 = 2x+y-2z --------(1)
-5 = 7x-y+z---------(2)
(1) + (2) gives us,
-2 = 9x-z ----------(3)

- anonymous

yep

- ganeshie8

thats ur solution line, every point on that line (3) is a solution.
which means, u can use any point on that line, that gives u linear combination of A B C which equals W

- ganeshie8

say, x =1,
put this in (3), and solve z
-2 = 9(1) - z
z = 11

- ganeshie8

put x=1, z=11 in (1) and solve y
3 = 2(1) +y-2(11)
y = 23

- ganeshie8

so, when x =1, we have y=23, z=11

- ganeshie8

check if this combination works

- ganeshie8

W = Ax+By+Cz

- anonymous

it workss

- ganeshie8

we could have taken x=0, that would have simplified the calculaiton, but its ok.. u see that all points on that line can be used right ?

- anonymous

im still ed how adding 1 and 2 gives you the line of intersection though?

- anonymous

confused*

- ganeshie8

ohk, thats simple, i just got rid of one variable by 'elimination method'

- anonymous

yeah

- ganeshie8

so, if i understand ur question correctly,
you are asking, why adding both the equations gives line of intersection of planes ?

- anonymous

yeah :)

- ganeshie8

thats really very good question:)
before i answer that, let me ask you a similar q, u knw that intersection of two lines is a point, right ?

- ganeshie8

for ex, take below two lines :-
x+y = 2
x+2y = 1

- anonymous

yep and the intersection of two panes is a line

- ganeshie8

since they're not parallel, they will meet at a point for sure.

- ganeshie8

now tell me, how u wud find that intersection point ?

- ganeshie8

tell me how to find intersecting point of below two lines,
x+y = 2
x+2y = 1

- anonymous

equal them to eachother

- ganeshie8

yes, why it works ?

- ganeshie8

by adding the two planes vertically, I did the same previously

- anonymous

oh

- ganeshie8

you're trying to figure out, why adding them vertically gives the line of intersection ?

- anonymous

yes

- ganeshie8

we can find answer to that, once we find answer to below :-
find intersection point of below two lines
x+y = 2
x+2y = 1

- ganeshie8

follow me closely if u can,
to find intersection point, lets take the first equation
x+y = 2

- ganeshie8

we can add/subtract same thing to both sides, right ? so lets add "x+2y" to both sides

- ganeshie8

*subtract actually..
x+y = 2
subtract "x+2y" from both sides
x+y - (x+2y) = 2 -(x+2y)

- anonymous

yep

- ganeshie8

simplify left side

- ganeshie8

notice that, till now, 2nd equation dint come into picture

- ganeshie8

x+y = 2
subtract "x+2y" from both sides
x+y - (x+2y) = 2 -(x+2y)
x+y -x-2y = 2-(x+2y)
-y = 2-(x+2y)

- anonymous

yep

- ganeshie8

Now, look at 2nd equation, x+2y=1, so put that value on right side. (this step actually determines the solution we get lies on second equation also)

- anonymous

ok

- ganeshie8

x+y = 2
subtract "x+2y" from both sides
x+y - (x+2y) = 2 -(x+2y)
x+y -x-2y = 2-(x+2y)
-y = 2-(x+2y)
from 2nd equation, x+2y=1
-y = 2-1
y = -1

- ganeshie8

since you knw y, u can find x
x = 3
so, (3, -1) lies on both sides

- anonymous

oohhh ok

- ganeshie8

if u are convinced (3,-1) lies on both lines, then u will understand elimination method.
in elimination method also, when we add equations vertically, exact same thing happens

- ganeshie8

so, you started linear algebra is it

- anonymous

yep.

- ganeshie8

i did this few months back gilbert strang's course... it was very good, if u get extra time, go thru lectures... they're very helpful :)

- ganeshie8

http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/

- anonymous

ooh okay thanks so much for your help!

- ganeshie8

np :)

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