Determine whether the set B is a basis for the space V. (a) B = {x² + x + 1, x² + 2, x + 3} and V is the space of polynomials of degree < 2. 17 [1 (b) В — and V is the space of all (2 × 2) real matrices.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Determine whether the set \( B \) is a basis for the space \( V \).**

(a) \( B = \{ x^2 + x + 1, x^2 + 2, x + 3 \} \) and \( V \) is the space of polynomials of degree \(\leq 2\).

(b) \( B = \left\{ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \right\} \) and \( V \) is the space of all \( (2 \times 2) \) real matrices.
Transcribed Image Text:**Determine whether the set \( B \) is a basis for the space \( V \).** (a) \( B = \{ x^2 + x + 1, x^2 + 2, x + 3 \} \) and \( V \) is the space of polynomials of degree \(\leq 2\). (b) \( B = \left\{ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \right\} \) and \( V \) is the space of all \( (2 \times 2) \) real matrices.
Expert Solution
Step 1

Definition of basis:- A set is said to be basis if it satisfy the following property

(i) Set is nonempty

(ii) The vectors in the set is linearly independent

(iii) The vectors spans the vector space.

(a) Given set β=x2+x+1,x2+2,x+3 and also given that the vector space is the set of all polynomial of degree 2

That is V=a0+a1x+a2x2:a0,a1,a2R

Now we have to show that set β forms a basis for V

(i) Set β is nonempty

(ii) Now check set β is linearly independent or not.

αx2+x+1+βx2+2+γx+3=0α+βx2+α+γx+α+2β+3γ=0

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