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Chapter 4: Q6E (page 199)

TRUE OR FALSE

6. if f1,..,fn is a basis of a linear space V, then any element of V can be written as a linear combination of f1,...,fn.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of basis

A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.

A set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.

For example,

The space of finite-degree polynomials has a basis with infinitely many elements

1,x,x2,.....

02

According to the definition

The given term is exactly the definition of the basis mentioned in the above part.

Hence, the statement is true.

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Most popular questions from this chapter

Show that if Tis a linear transformation from Vto W, thenT(0v)=0wwhererole="math" localid="1659425903549" 0vand0ware the neutral elements of Vand W, respectively. If T is an isomorphism, show that T-1(0w)=0v.

Find the transformation is linear and determine whether the transformation is an isomorphism.

Find the basis of all 2X2diagonal matrix,and determine its dimension.

Find the transformation is linear and determine whether the transformation is an isomorphism.

Question: Find the transformation is linear and determine whether they are isomorphism.
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