Basis and Dimension of a Vector Space - MathGPT Lesson 8

Hello everyone, in this lesson, we'll be discussing the basis and dimension of a vector space. A basis of a vector space is a set of vectors that, combined, can represent every vector in the space, meaning they are spanning and linearly independent. The number of vectors in a basis is called the dimension of the space.

In this session, we'll concentrate on the crucial concepts of basis and dimension in a vector space. We'll determine whether a given set of vectors forms a basis for a particular vector space and, if so, the dimension of that space.

Let's consider the vector space R², which we've discussed in our previous lessons, and look at the following sets of vectors in R². We'll need to determine several things:

Part A: Are the vectors in the set linearly independent? For our standard basis, the answer is yes.
Part B: Does the set span R²? Again, yes for the standard basis.
Part C: Does the set form a basis for R²? Yes.
Part D: If the set forms a basis for R², what is the dimension of R²? The dimension is two since there are two basis vectors.

Next, let's consider the set S2, which is { (1,1), (1,-1) }. These vectors are not scaled multiples of each other, indicating that they're pointing in unique directions, meaning they're linearly independent and span R². Since the dimension of R² is two, any set that spans R² with only two vectors forms a basis.

However, if a set contains a vector that is a scaled multiple of another, such as in set S3, these vectors are not linearly independent, and they don't span R². Thus, this set doesn't form a basis for R², making the dimensionality question inapplicable.

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