Linear Combinations of Vectors

For what value(s) of ‘a’ are the following vectors linearly dependent:

(1, 5, −2), (0, 6, a) and (3, 13, −3)?

• A.a = 3

• B.a = -3

• C.a = 9

• D.a = -9

Let {u, v, w, z} be independent vectors in a vector space V,then

• A.{u + v, v + w, w + z, z + u} spans V.

• B.{u + v, v + w, w + z, z + u} is independent.

• C.Span {u + v,v + w, w + z,z + u} is contained in span {u, v, w z}.

• D.{u + v, v + w, w + z, z + u} is a basis of V.

Let {v₁, v₂, … , v_n} be dependent, nonzero vectors in a vector space V,then

• A. There exists ij such that v_i = kv_j for some scalar k.

• B. {v₁} is dependent.

• C. Span {v₁, v₂, …, v_n} has dimension smaller than n.

Let {u, v, w} be an independent set in a vector space,then

• A. u is a linear combination of v and w.

• B. { u , v , u + v } is independent.

• C. au + bv + cw = 0 for some nonzero scalars a, b and c.

• D. { u , u + v , u + v + w } is independent.

Let V be a vector space, and let S be a subset of V . What does it mean when we say that W is closed under

scalar multiplication?

• A. Whenever X is in V and c is a scalar, then cx is in W.

• B. Whenever X is in W and c is a scalar, then cx is in V.

• C. Whenever X is in W and c is a scalar, then cx is in W

• D.Whenever X is in V and c is a scalar, then cx is in V.

Let V be a five-dimensional vector space, and let S be a subset of consisting of five vectors, Then S

• A. Must be linearly independent, but cannot span V.

• B. Can span V , but only if it is linearly independent, and vice versa.

• C.Must be a basis of V.

• Must be linearly dependent, and must span V.

Let V be a five-dimensional vector space, and let S be a subset of consisting of three vectors,Then S

• A. Cannot span V , but can be linearly independent or dependent.

• B. Must be linearly dependent, but may or may not span V .

• C. Must be linearly independent, but cannot span V .

• D. Must be linearly dependent, and must span V .

Let be V a vector space, and S let be a subset of V . What does it mean when we say that S span V ?

• A.Every vector V in has exactly one representation

  as a linear combination of vectors in S .

• B.Every vector V in can be expressed as a linear

  combination of vectors in S.

• C.The elements of are all distinct from each other

• None of the above

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

• A.Every element in V is a linear combination of

  elements in S .

• B.The only way to write 0 as a linear combination of elements of S is the zero combination .

• C.The number of elements of S is less than or equal to the

  dimension of V .

• None of the above

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

• A.Every element of S is a linear combination of

  other elements of S.

• B.There is a way to write 0 as a linear combination of elements of S other than the zero combination.

• C.The span of S has smaller dimension than the

  dimension of V .

• None of the above.