Show wether the vectors are linear independent and if they generate .
a)
My attempt : I don't understand anything about my teacher's notes. So I've read on wikipedia that in order to determine if some vectors are l.i. I could calculate the determinant of a matrix formed by them. That's what I did here and I found it to be different from , so the 3 vectors are l.i.
Now how can I show that they generate or not? I guess I must find a basis (but I don't know how to procede and what it is) and check if it generates .
Thanks in advance.
This is what I've done.
thus they are linear independent. Am I calculating it badly?
And about I asked it myself : is that possible that they generate for example ? And I thought not, because it would mean that 2 of them are linear dependent and the remaining vector is linear independent from the 2 firsts, but I don't know how to show this.
I probably did an error since icemanfan claims that they are linear dependent...
Sorry not to start it in a new thread, but the entire question states "If they are not l.i., characterize implicitly the subspace generated and give a basis of this subspace".
I've really no idea about how to do that. Maybe they generate , but I don't know how to show it...nor how to pick off a basis.
you can do both parts of your problem at the same time without even using determinant: first let then is in the subspace spanned by the columns of
if and only if the equation has a solution. so you need to consider the augmented matrix
now it's clear that the columns of are linearly dependent because the third row of the echelon form of is all 0. this answers the first part of your question. also it's clear that is
consistent iff which gives us hence so the subspace generated by the columns of is a plane which goes through
the origin and is spanned by the vectors clearly is a basis for the subspace. this answers the second part of your question.