Question

Linear Algebra Conceptual Questions

• If a subset of a vector space is NOT a subspace, what are the four things that could go wrong? How could you check to see which of these four properties aren’t true for the subset?

• Is it possible for two distinct eigenvectors to correspond to the same eigenvalue?

• Is it possible for two distinct eigenvalues to correspond to the same eigenvector?

• What is the minimum number of vectors required take to span R^n? If you have that many vectors, is your set guaranteed to span R^n? If not, what has to be true about the vectors in order for them to span R^n?

• What is the maximum number of vectors that can form a linearly independent subset of R^n?

Answer 1

1) If a subset of vector space is not subspace then atleast one of following property is not satisfied

i) 0 is not belonging to that subset

ii) Closure property is not satisfied for addition

iii) Scaler multiplaction is not in that subset

iv) Additive inverse of a element of subset is not in that subset

2) yes it true, but they are linearly dependent

3)No it is not true. If possible let we have two distinct eigen values a and b belonging to same eigen vector v, then we have

Tv=av and Tv=bv this implies ab=bv

then a=b, which is contradictoion. Hence the result

4) To span R^n we have required atleast n elements. But they span R^n only if the set contain n linearly independent elements.

5) n is the maximum number of vectors that can form a linearly independent subset of R^n