Question

Is b a linear combination of the columns A?

Answer 1

\[A= \left[\begin{matrix}1 & -1&2&1 \\ 2&-3&2&0\\-1&1&2&3\\-3&2&0&3\end{matrix}\right]\]
b=\[\left(\begin{matrix}2 \\ 3\\6\\9\end{matrix}\right)\]

Answer 2

Do I go about solving Ax=b??

Answer 3

that wud be a long process

Answer 4

observe that,

last column = 3rd column - 1st column

Answer 5

But what does that mean?

Answer 6

let me think a bit... wat that means

Answer 7

it wont tell us much, i think we need to reduce the augmented matrix, and see if it has a solution

Answer 8

augmented matrix

\[

\left[\begin{matrix}1 & -1&2&1&2 \\ 2&-3&2&0&3\\-1&1&2&3&6\\-3&2&0&3&9 \end{matrix}\right]
\]

Answer 9

The words "Reduced Row Echelon Form" should come to mind. Don't panic when a row disappears! It will be okay. Just keep going.

Answer 10

None of my rows disappeared! now I'm worried @tkhunny

Answer 11

that means the system is consistent, and we have solution. So it is possible to write b as linear combination of column vectors of A

Answer 12

Oh! So if I did have a row of zeros there would be a variable and b would not be a linear combination?

Answer 13

if u have all 0's, we can still have solutions (infinite)

but if we have a row like : 0 0 0 0 8
then its impossible to get a 8 out of 0s -- so we wont be having any solution. and we say the system is inconsistent

Answer 14

since the matrix we have has det=0, there will be some verctors, b, which we cant reach. let me quickly cook up some b, for which its not possible to linearly combine column vectors

Answer 15

oh ok!

Answer 16

below b cannot be linearly combined using our matrix column vectors, try and see :)

b = \(\left(\begin{matrix}2 \\ 3\\6\\7\end{matrix}\right) \)

Answer 17

Wait I don't see why, all that does is change one numbe

Answer 18

yes, wid that change in number, u will not have all 0's in ur last row in augmented matrix

Answer 19

this is the original b :-
http://www.wolframalpha.com/input/?i=row+echleon++%7B%7B1+%2C+-1%2C2%2C1%2C2+%7D%2C+%7B2%2C-3%2C2%2C0%2C3%7D%2C+%7B-1%2C1%2C2%2C3%2C6%7D%2C+%7B-3%2C2%2C0%2C3%2C9%7D%7D

Answer 20

this is wid the cookedup b :-
http://www.wolframalpha.com/input/?i=row+echleon++%7B%7B1+%2C+-1%2C2%2C1%2C2+%7D%2C+%7B2%2C-3%2C2%2C0%2C3%7D%2C+%7B-1%2C1%2C2%2C3%2C6%7D%2C+%7B-3%2C2%2C0%2C3%2C7%7D%7D

Answer 21

Okay I got to go back and do it because mine looked nothing like that haha. But for now I'm going to get some sleep, thanks for your help!

Answer 22

np :) have good sleep... trust wolfram... this is big matrix, manually doing is a pain !

Answer 23

The reduced row echelon form takes skill and care. You cannot hurry through it. Don't be afraid to write the WHOLE MATRIX at each stage.