Q18E Find the basis of all Pn, and de... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q18E (page 176)

Find the basis of all $P_{n}$ , and determine its dimension.

Short Answer

Expert verified

The dimension of $P_{n}$ is $n + 1$ which is spanned by role="math" localid="1659419285127" $Span \{1, t, t^{2}, . . ., t^{n}\}$ .

Step by step solution

Determine the span

Consider the set of all polynomial $P_{n}$ .

The set role="math" localid="1659419586773" ${1, t, t_{2}, . . ., t_{n}}$ is linear independent set of $V$ if there exist constant $a_{i} 鈭� R$ such that $a_{0} + a_{1} t_{1} + a_{2} t_{2} + . . . + a_{n} t_{n} = 0$ where $a_{0} = a_{1} = a_{2} = . . . = a_{n} = 0$ .

Any polynomial $f (t) 鈭� P_{n}$ is defined as follows.

$f (t) = b_{0} + b_{1} t + b_{2} t^{2} + . . . + b_{n} t^{n}$

Compare the polynomials

Compare the equations $b_{0} + b_{1} t + b_{2} t^{2} + . . . + b_{n} t^{n} = 0$ bot side as follows.

$\begin{array}{l} b_{0} + b_{1} t + b_{2} t^{2} + . . . + . + b_{n} t^{n} = 0 \\ b_{0} + b_{1} t + b_{2} t^{2} + . . . + . + b_{n} t^{n} = (0) + (0) t + (0) t^{2} + . . . + (0) t^{n} \\ b_{i} = 0 \end{array}$

By the definition of linear independence, the subset $\{1, t, t^{2}, . . ., t^{n}\}$ is L.I where are linear independent.

Therefore, the set of polynomial $P_{n}$ is spanned by $S p a n \{1, t, t^{2}, . . ., t^{n}\}$ .

Hence, the dimension of $A$ is and $n + 1$ spanned by $S p a n 1, t, t^{2}, . . ., t^{n} .$

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91影视

Find the basis of all $P_{n}$ , and determine its dimension.

Short Answer

Step by step solution

Determine the span

Compare the polynomials

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91影视

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

Short Answer

Step by step solution

Determine the span

Compare the polynomials

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the basis of all $P_{n}$ , and determine its dimension.