Linear Combinations of Vectors

• Option (a)

• Option (b)

• Option (c)

• Option (d)

The linear span L(S) of any subset S of a vector space V(F) is a .........of V(F).

In a vector space of dimension 5, a subset A containing 6 elements is generating set. Then

• A is Linearly independent also.

• There is one element in A which can be written as a linear combination of others.

• A contains zero element

• Can not say anything about the set A

If W is a subspace of a vector space V(F) and α, β ∈ W then for any scalar c ..........

• cα + β ∈ V

• cα + β ∈ W

• cα + β ∉ V

• cα + β ∉ W

Is the vectors (1, 2, 3), (4, 5, 6) and (7, 8, 9) span the vector

space ℝ3.

• yes

• no

Let V be a vector space, and let W be a subset of V. What does it mean when we say that is linearly independent?

• S is closed under both addition and scalar multiplication.

• The only way to write 0 as a linear combination of elements of S.

• All the elements of are distinct from each other.

• S has nullity zero.

Let V be a five-dimensional vector space, and let S be a subset of V which spans V . Then S

• Must have exactly five elements.

• Must consist of at least five elements.

• Must have infinitely many elements.

• Must have at most five elements.

Determine if the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9) are linearly independent.

• No, the vectors are collinear.

• Yes, the vectors are linearly independent.

• No, the vectors are orthogonal.

• No, the vectors are not linearly independent.

What is the dimension of the vector space spanned by the vectors (1, 2, 3) and (4, 5, 6)?

• 5

• 2

• 3

• 0

The value of k so that vectors (1,-1,3),(1,2,-2),(k,0,1) are linearly independent.

• 5/4
  

• 1/4

• 3/4

• 7/4