2. State \"True\" or \"False\" for each of the following statements. You do not

Question

2. State "True" or "False" for each of the following statements. You do not need to justify your answers. (a) There exists a vector space consisting of exactly 100 vectors. (c) In a vector space of dimension 3, any three vectors are linearly (d) In a vector space of dimension 3, any four vectors are linearly (b) There exists a vector space of dimension 100 independent. dependent. e) Any vector space of dimension 2 has exactly two subspaces. (f) Any vector space of dimension 2 has infinitely many subspaces. (g) Any vector space of dimension 3 can be expanded by four vec- tors. (h) Any vector space of dimension 3 can be expanded by two vec- tors (i) Three vectors are linearly dependent if and only if one of them can be written as a linear combination of the other two. Gi) The column space and row space of the same matrix A will have the same dimension.

Explanation / Answer

2.(a). False. All linear combinations of the basis vectors will be in the vector space.

(b). True. R100 is a vector space of dimension 100.

(c ). False. The vectors (1,0,0)T,(2,0,0)T and (3,0,0)T in R3 are linearly dependent.

(d). True. In a vector space of dimension 3, a maximum of 3 vectors only can be linearly independent.

(e).False. Span{(1,0)T}, span {(0,1)T}, and span { (1,0)T,(0,1)T} are all subspaces of R2. Every vector space is a subspace of itself.

(f).True.

(g).False. A maximum of 3 vectors only can be linearly independent in a vector space V of dimension 3. All linear combinations of the basis vectors will be in the vector space. It cannot be expanded by 4 vectors.

(h).False. For the same reason as in (g) above.

(i).True. This is the definition of linear dependence.

(j).True. The dimension of Row(A) or Col(A) for any matrix A is called the rank of A.

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

---

---

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