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Hello everyone, The problem says:Determine which, if, any, of the following sets span R^3?

b. {[1 1 -1], [-1 1 1]} note: i'm just transposing them, they should be verical.

c. {[1 1 0], [0 1 1], [1 0 0], [1 0 1]}

Those are the 2 sets, he showed us how to do them but i got lost on his steps:

he writes:

Intially b does not span R^3, which makes sense, because there are only 2 vector sets in b. He then goes on and writes

[1 1 -1] [-1 1 1] * [x y z];

and says use the dot product and name the first column which is, [1 1 -1] x1, then the 2nd column [-1 1 1], x2 and use the dot product.

he comes out with:

x+y-z = 0;

-x + y + z = 0;

he then writes:

y = a;

2y = 0;

y = 0;

x = z;

he totally lost me and now your probably lost as well. Anywho from that he got:

[1 0 1] which says will make b span R^3 and you would get:

{[1 1 -1], [-1 1 1], [1 0 1]}

Any ideas on how he got that last vector? Also i know if the dot porduct of 2 vectors is 0, then its linear indepdant.

He then says, create 3 bases for R^3 using the above sets. he showed us this:

[1 1 0] [0 1 1] [1 0 0] [1 0 1]

[1 0 1] = -[1 1 0] + [ 0 1 1] + 2[1 0 0]

so he said u don't need

[1 0 1], but he kind of did this by looking at it, is there a systematic approach to solving it? Thanks.

b. {[1 1 -1], [-1 1 1]} note: i'm just transposing them, they should be verical.

c. {[1 1 0], [0 1 1], [1 0 0], [1 0 1]}

Those are the 2 sets, he showed us how to do them but i got lost on his steps:

he writes:

Intially b does not span R^3, which makes sense, because there are only 2 vector sets in b. He then goes on and writes

[1 1 -1] [-1 1 1] * [x y z];

and says use the dot product and name the first column which is, [1 1 -1] x1, then the 2nd column [-1 1 1], x2 and use the dot product.

he comes out with:

x+y-z = 0;

-x + y + z = 0;

he then writes:

y = a;

2y = 0;

y = 0;

x = z;

he totally lost me and now your probably lost as well. Anywho from that he got:

[1 0 1] which says will make b span R^3 and you would get:

{[1 1 -1], [-1 1 1], [1 0 1]}

Any ideas on how he got that last vector? Also i know if the dot porduct of 2 vectors is 0, then its linear indepdant.

He then says, create 3 bases for R^3 using the above sets. he showed us this:

[1 1 0] [0 1 1] [1 0 0] [1 0 1]

[1 0 1] = -[1 1 0] + [ 0 1 1] + 2[1 0 0]

so he said u don't need

[1 0 1], but he kind of did this by looking at it, is there a systematic approach to solving it? Thanks.

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