Question

A question about linear algebra.... See attachment please

Answer 1

My question is...... are there times when you can't express the result as a linear combination of the two others?

Answer 2

If so is what clues you in on that?

Answer 3

Maybe lets try an example in \(xy\) plane first :

can you express the vector \((2,2)\) using a linear combination of vectors \((1,0)\) and \((2,0)\) ?

Answer 4

So when you say the vector (2,2) that would be \[a_{1}=\left(\begin{matrix}2 \\ 2\end{matrix}\right)\] correct?

Answer 5

right

Answer 6

ok give me a second to try it out

Answer 7

take ur time

Answer 8

I can see it now because both of the "Y" elements are zero then there is no multiple of zero that could combine to equal 2

Answer 9

Thank you, is there a method in general (by inspection) that I can use to check one of these before attempting to work the problem out?

Answer 10

so what do you conclude ?

Answer 11

Yes, there is a method. Before getting to that, I just want to see you get the idea of taking linear combinations of vectors..

Answer 12

If all X,Y,Z, or Nth element of each vector is zero and the resultant vector is non-zero I can concluded that there is no linear combination of the vectors that will give the correct resultant vector

Answer 13

And I am not sure i am getting the idea your referring to

Answer 14

I am not referring to any idea yet

Answer 15

The present problem is cooked up to be done by visual inspection

Answer 16

What I meant is you said " I just want to see you get the idea of taking linear combinations of vectors." and honestly I am not sure that I am

Answer 17

for part (i), try \(3a_1 + 2a_2\)

Answer 18

for part (ii), try \(3a_1 + 4a_2\)

Answer 19

I actually found a solution for both....

Answer 20

So I was trying to figure out 1) if there was a time this wouldn't work. (which you have shown me) and a way to inspect them and get a definite "NO" sometimes.... if what I am saying makes sense