Determine if the statements below are correct or not a For a

Determine if the statements below are correct or not

a) For a given vector space, there must be a unique basis. In other words, there can never be multiple sets of basis vectors for the same vector space.

b) If there is a set of linearly dependent vectors, then any one vector from this set can be expressed as a linear combination of the remaining vectors in the set.

Solution

(a) STATEMENT: NOT CORRECT

BASIS: A set of vectors in a vector space V is called a basis, if the vectors are linearly independent and every vector in the vector space is a linear combination of the vectors in this set. Thus, a basis is a linearly independent spanning set.

Therefore, a vector space can have many distinct sets of basis vectors, but each of the set has the same number of elements in it.

(b) STATEMENT: NOT CORRECT

A finite set V= {x₁, x₂, . . . , x_m} of vectors in Rⁿ is said to be linearly dependent if there exist scalars (real numbers) c₁, c₂, . . . , c_m, not all of which are 0, such that c₁x₁ + c₂x₂ + . . . + c_mx_m = 0.

  If a set of vectors is linearly dependent, then one of them can be written as a linear combination of the others, provided its coefficient is not zero.