Find a basis for span(l - x, x - x2, 1 - x2, 1 - 2x + x2) in P 2

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 11CM

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Question

Find a basis for span(l - x, x - x², 1 - x², 1 - 2x + x²) in P 2

Expert Solution

Step 1

We know that $P_{n}$ is the vector space of all polynomials of degree less than or equal to n. This implies

that $P_{2}$ is the vector space of all polynomials of degree less than or equal to 2.

The basis of the vector space $P_{2}$ is the set $B = \{1, x, x^{2}\}$ . We have to find the basis for the span of the

set $\{1 - x, x - x^{2}, 1 - x^{2}, 1 - 2 x + x^{2}\}$ .

Step 2

For the polynomial $1 - x$ , we can write $[\begin{matrix} 1 & x & x^{2} \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}]$ . For the polynomial $x - x^{2}$ , we can write

$[\begin{matrix} 1 & x & x^{2} \end{matrix}] [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]$ .

For the polynomial $1 - x^{2}$ , we can write $[\begin{matrix} 1 & x & x^{2} \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}]$ . For the polynomial $1 - 2 x + x^{2}$ , we can write

$[\begin{matrix} 1 & x & x^{2} \end{matrix}] [\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}]$ . So we can write the vectors in the matrix form as $[\begin{matrix} 1 & 0 & 1 & 1 \\ - 1 & 1 & 0 & - 2 \\ 0 & - 1 & - 1 & 1 \end{matrix}]$ .

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