Coordinate vector basis proving question

[gtse]

Homework Statement

Let S = {v1,v2,...,vn} be a basis for an n-dimensional vector space V. Show that {[v1]s,[v2]s,...[vn]s} is a basis for Rn.
Here [v]s means the coordinate vector with respect to the basis S.

Homework Equations

[v]s is the coordinate vector with respect to the basis S.

The Attempt at a Solution

S={v1..vn} is a basis and must be linearly independent.
Any vector v in S then is a unique linear combination of the vectors in S, so v=a1v1+a2v2+...+anvn.
Since [v]s in general = (a1,a2,...an), then every [vi]s where i = 1 .. n has a unique (a1,a2,...,an) and so the basis {[v1]s,...,[v2]s} will be linearly independent and thus form a basis for Rn.


I have no answers to verify with, so I would like to know if I have answered it correctly. I am extremely weak with anything to do with proving so any assistance would be greatly appreciated, :).

 

[mighty2000]

I would approach this problem by first understanding the definitions and concepts involved. A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a linear combination of the basis vectors.

Next, I would consider the concept of coordinate vectors with respect to a basis. This means that for any vector v in the vector space, [v]s represents the unique set of coordinates that describe v with respect to the basis S.

Now, let's look at the set {[v1]s,[v2]s,...[vn]s}. This set contains the coordinate vectors of the basis vectors v1, v2, ..., vn with respect to the basis S. Since S is a basis for V, this means that any vector in V can be written as a linear combination of the basis vectors (i.e. v=a1v1+a2v2+...+anvn). And since [v]s represents the coordinates of v with respect to S, this means that [v]s can also be written as a linear combination of [v1]s, [v2]s, ..., [vn]s.

Therefore, the set {[v1]s,[v2]s,...[vn]s} spans Rn, since any vector in Rn can be written as a linear combination of these coordinate vectors. And since these coordinate vectors are also linearly independent (since S is linearly independent), this set forms a basis for Rn.

In conclusion, the set {[v1]s,[v2]s,...[vn]s} is a basis for Rn because it spans Rn and is linearly independent. This can be proven by using the definitions and concepts of basis and coordinate vectors.