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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 set of all polynomial f(t) in P2 such that f(1)=0, and determine its dimension.

The dimension of P2 such that f1=0 is which is spanned by Span{(t-1),(t2-1)}.

See the step by step solution

Step by Step Solution

Step 1: Determine the span set.

Consider the set of all polynomial such that .

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

Any polynomial f(t)P2 is defined as follows.


Substitute the value 1 for t in the equation f(t)=b0+b1t+b2t2 as follows.


As f(1)=0, substitute the value 0 for f1 in the equation f(1)=b0+b1+b2 as follows.

f1=b0+b1+b2 b0=-b1+b2

Substitute the value -b1+b2for in the equation f(t)=b0+b1t+b2t2as follows.


From the equation, the set (t-1),(t2-1)span P2 if f(1)=0.

Step 2: Determine the property of independency.

Compare the equations b1(t-1)+b2(t2-1)=0 both side as follows.

b1t-1+b2t2-1=0b1t-1+b2t2-1=0t-1+0t2-1 bi=0

By the definition of linear independence, the subset (t-1),(t2-1)is L.I.

Therefore, the set of all polynomial P2 such that f1=0 has dimension 2 and spanned by Span(t-1),(t2-1).

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