Vectors span R^3

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March 13th 2009, 11:40 AM
antman

Vectors span R^3

Which of the following sets of vectors span R^3?
a. {(1,-1,2),(0,1,1)}
b. {(1,2,-1),(6,3,0),(4,-1,2),(2,-5,4)}
c. {(2,2,3),(-1,-2,1),(0,1,0)}
d. {(1,0,0),(0,1,0),(0,0,1),(1,1,1)}
March 13th 2009, 12:18 PM
Ruun

Hi!

I'm spanish so I hope that span means... ehm "make" $\mathbb{R}^3$ . If you have a vector space for example $\mathbb{R}^n$ , then the dimension is n. The dimension of a vector, as far as I know, it's a maximal set of linear independent vectors or a minimal set of "generator" vectors. In practise it's the number of vector that every base should have. In our case as $n=3$ then the dimension is $3$ .

With this in mind, option 1) doesn't have 3 vectors so it can't span $\mathbb{R}^3$ . 2) Four vectors, it can be a good candidate. If there are 3 or more linearly independent then it's ok. This is the math around this.

To proof if they are or not linearly independent, just look if there is a determinant of a matrix that have as rows or columns your vectors, different of zero.

Hope it helps (Hi)
March 21st 2009, 06:45 AM
antman

I tried to follow examples of how to solve this problem in my textbook, but none of them were in the same format.

For part a, my work is:

http://www.mathhelpforum.com/math-he...696b683f-1.gif

http://www.mathhelpforum.com/math-he...b130a86b-1.gif
http://www.mathhelpforum.com/math-he...b1c55ba0-1.gif
http://www.mathhelpforum.com/math-he...3f2e0eac-1.gif

http://www.mathhelpforum.com/math-he...d7f90509-1.gif

Reduced
http://www.mathhelpforum.com/math-he...30a34c21-1.gif

The system is inconsistent and has no solution so this set does not span R^3?

Part b: same work resulting in the following reduced matrix

http://www.mathhelpforum.com/math-he...68ddc293-1.gif
So this would also not span R^3 since there are no solutions?

Part c:

http://www.mathhelpforum.com/math-he...270e4494-1.gif
http://www.mathhelpforum.com/math-he...b098f00e-1.gif
http://www.mathhelpforum.com/math-he...9fab436f-1.gif
So this set does span R^3?

Does anyone know if this is correct? Thank you!
March 21st 2009, 08:10 AM
HallsofIvy

You posted this same thing under "urgent homework help" and I responded there. Didn't you notice where it said "Don't make a duplicate thread elsewhere, please"?