Subspaces and Basis of vector spaces

[goldfronts1]

I am totally lost on the following questions. What does exhibit mean?

1) Show that the given set H is a subspace of ℜ^3 by finding a matrix A such
that N(A) = H (in this case, N(A) represents the null space of A).

2) Exhibit a basis for the vector space H.

a
b {for all R^3: a=b-c and 2a-b=b-c}
c


3) Show that the given set W is a subspace of ℜ^4 by finding a matrix B such
that Col(B) = W (in this case, Col(B) represents the column space of B).

4) Exhibit a basis for the vector space W.

a-3b+c
2b-11c
a-3b+9c {for all R^4: a,b,c for all R }
c+a-b


Any help is greatly appreciated

 

[Hurkyl]

What does exhibit mean?

It means "find one, and prove that it is one".

 

[goldfronts1]

I am totally lost on how to find a basis for a vector space.

On #3 I found matrix B, but am confused on how to find a basis for the vector space?

On #1 and #2 I am totally confused.

 

[goldfronts1]

Basis for Vector Space and Subspaces

Can someone please explain how to find a basis for a vector space?

Thanks

 

[radou]

Can someone please explain how to find a basis for a vector space?

Thanks

Example: if you have a set W = {(a, b, c) in R^3 : a + b - c = 0}, simply write (a, b, c) = (- b + c, b, c) = b(-1, 1, 0) + c(1, 0, 1), where b and c are in R. Obviously, the set {(-1, 1, 0), (1, 0, 1)} spans your subspace W. What condition needs to be satisfied for this set to form a basis for W?

 

[goldfronts1]

That,

a+b-c=0

 

[radou]

That,

a+b-c=0

You better start thinking what you're talking about.

 

[goldfronts1]

What are you talking about? If you want to make fun of people that are confused then that is not cool. I told you I am lost and need your help.

Can you please explain what you are trying to say?

 

[radou]

If you want to make fun of people that are confused then that is not cool.

I am not making fun of you.

Can you please explain what you are trying to say?

What is the definition of a basis?

 

[goldfronts1]

This is the textbook definition:
a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others.

But what that exactly means, I'm not sure.

 

[HallsofIvy]

I am totally lost on the following questions. What does exhibit mean?

Strictly speaking "exhibit" means "show". Of course, to "show" a basis, you will need to find it as Hurkyl said!

1) Show that the given set H is a subspace of ℜ^3 by finding a matrix A such
that N(A) = H (in this case, N(A) represents the null space of A).

Is the "given set H" the same as in exercise 2?

2) Exhibit a basis for the vector space H.

a
b {for all R^3: a=b-c and 2a-b=b-c}
c

You have two equations in 3 variables. Unless they happen to be dependent, you can solve for two of the variables in terms of the third. That means the subspace will be one dimensional. (Geometrically, each of those equations is the equation of a plane containing the origin. Their intersection is a line through the orgin- a one-dimensional subspace of R³. In particular, you can always take that "third" variable to be equal to 1. If you take c= 1 and solve for a and b, what do you get? Now put those 3 values into your vector.
Oops! They are dependent. If you set c= 1, you get a= b-1 and 2a= 2b-1 which cannot be solved for a and b! The subspace is actually 2 dimnsional. Take b= 1, c= 0. What is a? Now take b= 0, c= 1. What is a? Show that those two vectors for a basis for the space. (You don't really need to use 0 and 1 but those numbers are simplest.)

3) Show that the given set W is a subspace of ℜ^4 by finding a matrix B such
that Col(B) = W (in this case, Col(B) represents the column space of B).

Again, is W the same as in 4?

4) Exhibit a basis for the vector space W.

a-3b+c
2b-11c
a-3b+9c {for all R^4: a,b,c for all R }
c+a-b


Any help is greatly appreciated

Since you can choose a, b, c to be any 3 numbers, that will be three dimensional. A good way to handle this is to let each of a, b, c be equal to 1 while the others are 0. In other words, if a= 1, b= c= 0, what vector do you get? If b= 1, a= c= 0, what vector do you get? If c= 1, a= b= 0, what vector do you get. Can you PROVE that those three vectors for a basis for the subspace. (You will need to use the definition of "basis" to do that.)

 

[goldfronts1]

Yes, for Question 1, H is the same H in question 2.

And in Question 3, W is the same W in question 4.