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

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

Mathematics
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My question is...... are there times when you can't express the result as a linear combination of the two others?
If so is what clues you in on that?

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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 \((2,0)\) ?
So when you say the vector (2,2) that would be \[a_{1}=\left(\begin{matrix}2 \\ 2\end{matrix}\right)\] correct?
right
ok give me a second to try it out
take ur time
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
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?
so what do you conclude ?
Yes, there is a method. Before getting to that, I just want to see you get the idea of taking linear combinations of vectors..
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
And I am not sure i am getting the idea your referring to
I am not referring to any idea yet
The present problem is cooked up to be done by visual inspection
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
for part (i), try \(3a_1 + 2a_2\)
for part (ii), try \(3a_1 + 4a_2\)
I actually found a solution for both....
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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

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