Question

let S={V1,V2,V3} where V1=(1,2,3),V2=(2,1,4),V3=(3,0,5) then find a basis for the subspace V=span of R3(3-dimension)

Answer 1

setup a matrix and row-reduce

Answer 2

yah i tried that and it is giving me 0=0 in the last row

Answer 3

yeah, so which vector can you add to span \(\mathbb R^3\) ?

Answer 4

http://www.wolframalpha.com/input/?i=row%20reduce%20%7B%5B1%2C2%2C3%5D%2C%5B2%2C1%2C4%5D%2C%5B3%2C0%2C5%5D%7D&t=crmtb01
we can add a standard basis vector in place of the third vector and that should cure our problem

Answer 5

meaning V3 is not in the basis?

Answer 6

ohh i mean V1 and V2 are not in the basis?

Answer 7

5x+2y-3z = 0
1 2 3
--------------
5 + 4 - 9 =0


5x+2y-3z = 0
2 1 4
--------------
10+2- 12 =0


5x+2y-3z = 0
3 0 5
--------------
15+0 -15 =0

Answer 8

then how do we conclude ?

Answer 9

so, a basis is a ... efficient span, if i recall this stuff correctly, right?

Answer 10

since v3 can be created from v1 and v2, we can ignore v3 is what i believe the row reducing tells us; and that the nullspace can be constructed by using the free variable

Answer 11

according to my view ,V2 and V1 are not in the basis since they are dependent. tell me if im wrong

Answer 12

We have a span of 3 vectors V1, V2, and V3; to form a basis, we need to weed out the dependant ones (those that can be formed from one or more vectors in the set). The row reducing process tells us which of these vectors are "free variables" and can be ignored.

So the basis for the subspace is B={V1,V2}

Answer 13

ok,so what if they say express each vector not in the basis as a linear combination of the found basis vectors.how are we gonna atempt it

Answer 14

the row reducing tells us that

V3 = 5/3 V1 + 2/3 V2 + 0V3 does that sound right?

Answer 15

oww i see, since v3 is not in the basis right?

Answer 16

correct, but im going off of Turings posted, which to me looks more like a row space

mine is: http://www.wolframalpha.com/input/?i=rref+%7B%7B1%2C2%2C3%7D%2C%7B2%2C1%2C0%7D%2C%7B3%2C4%2C5%7D%7D

Answer 17

- v1 = -1,-2,-3
2 v2 = 4, 2, 8
--------------
v3 = 3, 0 ,5

Answer 18

since v3 can be created from the other 2vectors, there is no need to include it in a basis (a most efficient span)

Answer 19

wow, thanks #happy#.........if i may ask, what efficient span simply mean

Answer 20

efficiency means that we use only the things we have to use.

a "span" can include any set of vectors, regardless of their state of dependance or independance.

a "basis" is that span with all the useless vectors removed so that we only have a set of vectors that can produce all the rest of them.

Answer 21

mmmmmmmm thanks

Answer 22

youre welcome