Question

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)

Answer 1

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]
\]

Answer 2

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

Answer 3

yep

Answer 4

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

Answer 5

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

Answer 6

W = Ax + By + Cz

you need to find x, y, z

Answer 7

so yes, two equations, and 3 unknowns

Answer 8

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

Answer 9

you will get infinite solutions, just pick one ?

Answer 10

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

Answer 11

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

Answer 12

yep

Answer 13

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

Answer 14

a line has infinite points. so infinite solutions

Answer 15

lets find them..

Answer 16

few of them atleast :)

Answer 17

okay

Answer 18

:)

Answer 19

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


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

Answer 20

yep

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

check if this combination works

Answer 26

W = Ax+By+Cz

Answer 27

it workss

Answer 28

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 ?

Answer 29

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

Answer 30

confused*

Answer 31

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

Answer 32

yeah

Answer 33

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

Answer 34

yeah :)

Answer 35

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 ?

Answer 36

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

Answer 37

yep and the intersection of two panes is a line

Answer 38

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

Answer 39

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

Answer 40

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

Answer 41

equal them to eachother

Answer 42

yes, why it works ?

Answer 43

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

Answer 44

oh

Answer 45

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

Answer 46

yes

Answer 47

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

Answer 48

follow me closely if u can,

to find intersection point, lets take the first equation
x+y = 2

Answer 49

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

Answer 50

*subtract actually..

x+y = 2

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

Answer 51

yep

Answer 52

simplify left side

Answer 53

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

Answer 54

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)

Answer 55

yep

Answer 56

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)

Answer 57

ok

Answer 58

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

Answer 59

since you knw y, u can find x
x = 3

so, (3, -1) lies on both sides

Answer 60

oohhh ok

Answer 61

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

Answer 62

so, you started linear algebra is it

Answer 63

yep.

Answer 64

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 :)

Answer 65

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

Answer 66

ooh okay thanks so much for your help!

Answer 67

np :)