True/False a. T F There is a vector space consisting of exactly two vectors. b.

Question

True/False a. T F There is a vector space consisting of exactly two vectors. b. T F If S is a set of four vectors in P_4, then S cannot be a basis of P_4. c. T F The line y = x + 1 is a subspace of R^2. d. T F If T: V rightarrow W is a linear transformation, then the kernel is a subspace of V. e. T F Suppose T: V rightarrow W is a linear transformation from an n-dimensional space V to an m-dimensional space W, then rank(T) + nullity(T) = m. f. T F By appropriately using coordinate vectors, it is possible to any linear transformation as a matrix multiplication.

Explanation / Answer

9.

a) True. Z2 = {0,1} is an example, It is a finite field with just 2 elements in it. Any field is a vector space over itself.

b) True. Simply a set cannot form a basis for any of the vector spaces. A basis is a 'linearly independent' set of vectors, capable of spanning the whole space. So in general, any given set of 4 vectors cannot be a basis for P4.

c) False. The equation of the line does not satisfy the point (0,0). A subspace must necessarily have the (0,0) element. Hence y = x + 1 cannot be a subspace of R2, but straight lines passing through origin of the form y = mx, forms a subspace of R2

d) True. Kernel(T) is a subspace of V, the domain and the range (T), a subspace of W, the co domain.

e) False. Rank(T) + Nullity(T) = dim (V) not dim(W)

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.