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# Vectors - basis question watch

1. I understand that a basis is a linearly independent set of vectors which spans a subspace of ... but I really have no idea where to even begin with this question. Help please?

If {} is a basis for then prove that every vector in can be expressed in the form: in exactly one way where for all .
2. since spans every vector in is expressible as a linear combination of the vectors in

suppose you take an arbitrary vector in which can be written as:

but also as:

subtract one from the other and you get equality...
3. (Original post by Hasufel)
since spans every vector in is expressible as a linear combination of the vectors in

suppose you take an arbitrary vector in which can be written as:

but also as:

subtract one from the other and you get equality...

Oh gosh... this makes so much sense!!
I think I must have just been having a mental block. Thank you so much
4. (Original post by Hasufel)
since spans every vector in is expressible as a linear combination of the vectors in

suppose you take an arbitrary vector in which can be written as:

but also as:

subtract one from the other and you get equality...

After going through this I've actually become a bit stuck.

I have:

and

I subtract them:

... and I get confused about the coefficients here.

Are and the same number? ... that would make sense
5. (Original post by SummerPi)
After going through this I've actually become a bit stuck.

I have:

and

I subtract them:

... and I get confused about the coefficients here.

Are and the same number? ... that would make sense
since you have a zero vector, all the coefficients must be zero (linear independence) which means:

etc - i.e. the coeffs are equal, so you`ve demonstrated that the vector is unique

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