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Found in: Page 199

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# TRUE OR FALSE6. if ${{\mathbit{f}}}_{{\mathbf{1}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{{\mathbit{f}}}_{{\mathbf{n}}}$ is a basis of a linear space V, then any element of V can be written as a linear combination of ${{\mathbit{f}}}_{{\mathbf{1}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{{\mathbit{f}}}_{{\mathbf{n}}}$.

The given statement is true.

See the step by step solution

## 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

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

## 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.