If V is a finite-dimensional vector space, then a bilinear form f on Vis symmetric if and only if its matrix A in some (or every) ordered basis is symmetric.

Find a basis for the span of {(1,1,0, 1), (0, 1, 1, 0), (1, 0, 1, 1), (1, 0, 1,0)} in the vector space F. What is its dimension? Give an example of an infinite dimensional vector space over R.

Let S be a linearly independent set of vector form a finite dimensional vector space V. Prove that there exists a basis of V containing S.

Which of the following statements is (are) true if B is a basis for the vector space V (a) In some situations, a plane in R6 can be isomorphic to R4 (b) If x is in V and B contains n vectors, the the B- coordinate vector of x is in R (c) The vector spaces P5 and RS are isomorphic. (d) If PB is the change of coordinates matrix, then [x]B = PB x for x in V (e) A coordinate mapping is a bijective mapping O All of the above Noneoftheabove Options a, b, c and e Options a, b and e Options b, c and e Options a, b, d and e Options b, c, d and e Options a, and b