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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 a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2(see Example 16)? Find a basis for those that are subspace,


{pt:01ptdt=0} is a subset and a subspace of P2.

See the step by step solution

Step by Step Solution

Step 1: Definition of subspace

A subset W of a linear space V is called a subspace of V if

(a) W contains the neutral element 0 of V.

(b) W is closed under addition (if f and g are in W then so is f+g )

(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W ).

we can summarize parts b and c by saying that W is closed under linear combinations.

Step 2: Application of given conditions on vector space V.

General expression for a quadratic polynomial with one variable is p(t)=at2+bt+c

Given 01ptdt=0.

From given,



01at2+bt+c=at33+bt22+ct01 =a3+b2+c c=-a3+b2 Therefore, pt=at2+bt-a3+b2.

Assume, qt=et2+dt-e3-d2..

Also, let us assume two scalars α , β,

Then, we can write as:

αpt+βqt=αat2+bt-a3-b2+βet2+dt-e3-d2 =αat2+αbt-a3α-b2α+βet2+βdt-e3β-d2β =t2αa+βe+tαb+βd-13αa+βe-12αb+βd

Therefore, the given set is closed under linear combinations. Hence, it is a subspace of P2.

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