# Question about vectors and dimensions ?

• Dec 8th 2010, 07:18 AM
abrahamtim
Question about vectors and dimensions ?
Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?

2) What would be the dimension of V in this case ?

3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
(3,-2,4) (1,-1,0) (3,-1,8)

4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3

5) Find the components of the vector (3,4,5) in the basis considered in 4

this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
• Dec 9th 2010, 05:19 AM
CaptainBlack
Quote:

Originally Posted by abrahamtim
Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?

They would need to be linearly independednt and span the space.

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2) What would be the dimension of V in this case ?
Well what do your notes say?

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3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
(3,-2,4) (1,-1,0) (3,-1,8)
Well are they linearly independednt? If no what is the largest subset of them which is linearly independednt?

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4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3
You know what you have to do, show they are linearly independednt and that they span R^3

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5) Find the components of the vector (3,4,5) in the basis considered in 4

this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
CB
• Dec 10th 2010, 05:04 AM
HallsofIvy
Quote:

Originally Posted by abrahamtim
Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?

2) What would be the dimension of V in this case ?

These two are basic definitions- they should be the first thing you learn. The dimension of a space is defined as the number of vectors in a basis.

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3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
(3,-2,4) (1,-1,0) (3,-1,8)
As Captain Black said, are they independent?

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4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3
Again, are they independent? Do they span $R^3$.

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5) Find the components of the vector (3,4,5) in the basis considered in 4
In other words you want to find numbers, a, b, c, such that a(1, 1, 0)+ b(1, 0, 1)+ c(0, 1, 1)= (3, 4, 5). That is, you must have a+ b= 3, a+ c= 4, b+ c= 5. Solve those equations.

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this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
To revise? If you are reviewing a course you have already taken, you certainly should know basic definitions!