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Linear Algebra With Applications
Found in: Page 176
Linear Algebra With Applications

Linear Algebra With Applications

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

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Short Answer

Find the basis of all Pn, and determine its dimension.

The dimension of Pn is n+1 which is spanned by role="math" localid="1659419285127" Span1,t,t2,...,tn.

See the step by step solution

Step by Step Solution

Step 1: Determine the span

Consider the set of all polynomial Pn.

The set role="math" localid="1659419586773" {1,t,t2,...,tn}is linear independent set of V if there exist constant aiR such that a0+a1t1+a2t2+...+antn=0 where a0=a1=a2=...=an=0 .

Any polynomial ftPn is defined as follows.


Step 2: Compare the polynomials

Compare the equationsb0+b1t+b2t2+...+bntn=0 bot side as follows.


By the definition of linear independence, the subset 1,t,t2,...,tn is L.I where are linear independent.

Therefore, the set of polynomial Pn is spanned by Span1,t,t2,...,tn .

Hence, the dimension ofA is and n+1 spanned bySpan1,t,t2,...,tn.

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