Short Answer

TRUE OR FALSE
6. if $f_{1}, . ., f_{n}$ is a basis of a linear space V, then any element of V can be written as a linear combination of $f_{1}, . . ., f_{n}$ .

The given statement is true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: 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

$\{1, x, x^{2}, . . . .\}$ .

Step 2: 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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Short Answer

TRUE OR FALSE
6. if $f_{1}, . ., f_{n}$ is a basis of a linear space V, then any element of V can be written as a linear combination of $f_{1}, . . ., f_{n}$ .

Step by Step Solution

Step 1: Definition of basis

Step 2: According to the definition

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Linear Algebra With Applications

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TRUE OR FALSE6. 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.

Step by Step Solution

Step 1: Definition of basis

Step 2: According to the definition

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TRUE OR FALSE
6. if $f_{1}, . ., f_{n}$ is a basis of a linear space V, then any element of V can be written as a linear combination of $f_{1}, . . ., f_{n}$ .