Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given by p1 (x) = 2, p2(x) = x + 2x², and p3(x) = 6x + aa². a) (You are free to assume that the functions 1, x and x? are linearly independent.) b) combination of p1, P2 Find the condition on a e R that ensures that {P1, P2, P3} is a basis for P2. In the case that a = 6, write the function p(x) = 2 – x + x2 as a linear and P3.

Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given by p1 (x) = 2, p2(x) = x + 2x², and p3(x) = 6x + aa². a) (You are free to assume that the functions 1, x and x? are linearly independent.) b) combination of p1, P2 Find the condition on a e R that ensures that {P1, P2, P3} is a basis for P2. In the case that a = 6, write the function p(x) = 2 – x + x2 as a linear and P3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Q2. Use a scalar product the check whether the points A (0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2,…

Q: let v = (1, 3, −2) and w = (2, −1, 4). Compute v × w.

Q: Identify the maximum value and the corresponding unite vector of the quadratic form Q = x1 3a²³ —…

Q: 1. Find an orthogonal change of variables that eliminates the cross product terms in the quadratic…

A: Given Q=x12-4x1x2+x22

Q: Show that V1, V2, V3 are linearly independent, but V₁, V2, V3, V4 are not. Then express V4 as a…

Q: Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the…

A: It is given that r1(t) and r2(t) are vector valued functions in 2-space. The objective is to explain…

Q: 2. Is the following set of polynomials linearly independent in P2? Explain your work. (-1+2t + 3t?),…

Q: ) Let B = {vi =,₂ =, V3 =}. Determine whether B is linearly independent or linearly dependent.

A: Let us consider a set of vectors V=v1,v2,.......vn. The set of vectors are said to be linearly…

Q: show that the given vector functions are linearly dependent on (−∞,∞).

Q: (a) [1 2]. [–1 1] (c) [1 3], [2 – 3]. [0 2] (d) [2 4], [-1 2] (b) [0 0]. [1 1]. [–2 – 2]

Q: Find the unique quadratic polynomial y = ax² +bx+c that passes through the three points (1, 1),…

A: What is Polynomial: A polynomial is a real valued function. It is an algebraic function that is…

Q: 5а+ 2b Decide whether the set of all vectors of the form is a subspace of R³ or a

Q: 3. Check whether (2, 4) and (1, 2) are linearly dependent or not?

A: Linearly department vectors

Q: Are the polynomials x? + 1,x? – 1, 2x – 1 linearly independent in P2?

Q: Is A = diagonalizable. Justify your conclusion. 10 120 -3 5 2,

Q: Let A = -1 -3 3 and b = b₁ 8} Show that the equation Ax=b does not have a solution for some choices…

Q: Suppose |||| = 5 and |||| = 6, and assume and y are orthogonal. Then ||-||²7

Q: All computations must be shown. 1. Let pi(t) = 1+ t², p2(t) = t +t² and p3(t) = 1 + 2t + t² be…

A: Consider the provided question, given,p1t=1+t2p2t=t+t2and p3t=1+2t+t2

Q: Find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject…

Q: Find the length of the projection of [−8 −7]onto the vector [−7 5] Its written vertically…

Q: Which of the following operators in R'are linear? 1. T(x) = (8x2, -5x, -9x) %3D 2 T(x) = (10x; - 7x2…

Q: I have performed the following polynomial multiplication in matlab on two vectors of polynomial…

A: Since matlab is mymult=conv(v1, v2) and result is mymult=3 45 -6 -78 99 65 -54 -12 -10 240

Q: If V1 = [1 4 0] and V2 = [ -2 0 8], then determine all possible 1x3 row vectors V3 such that the set…

Q: Calculate the 1st order Newton polynomial that interpolates the two data points (1,1),(2,3). Now…

A: We can use Newton's forward difference interpolation method to find Newton's polynomial.

Q: Q7) True or False, and Justify your answer. If v1, v2, ... V4 are in Rª and v3 = 2v1 + v2, then…

Q: (b) Are the polynomials x + 5, x – 5, 2x +1 linearly independent in P1(R)? Explain.

A: The vectors a1, ..., an are called linearly independent if there are no non-trivial combination of…

Q: 2. (a) Define the dot product of two polynomials p(x) = ao+a₁x+a₂x²+a3x³ and q(x) = bo+b₁x+b₂x²+b3x³…

Q: X b) Let T 10 Find the Kernel of T. Write down the range of T as a span of independent 5x vectors.…

A: Obtain Kernel of T as follows.

Q: T: P2 P₂ defined by T(a + bx + cx2) = a + b(x + 8) + b(x + 8)² Find the following images of the…

Q: Prove that in P2 (the set of all polynomials with degree n< 2), every set with more than three…

A: Linearly dependent set: A set of vectors α1, α2,α3, … ,αn is said to be linearly dependent if ∃ c1,…

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be
given by p1(x) = 2, p2(x) = x + 2x², and p3(x) = 6x + ax².
а)
(You are free to assume that the functions 1, x and x? are linearly independent.)
b)
combination of p1, P2 and p3.
%3D
Find the condition on a e R that ensures that {P1, P2, P3} is a basis for P2.
In the case that a =
6, write the function p(x) = 2 – x + x² as a linear
%D

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Compute the quadratic form x Ax for A= Xx₁ 12 - a. x = b. x= 1 X2 a. x¹Ax= 3 2 [1] 10 C. X = and each of the following vectors.
which of the pollowing rets functions that contain are linearly independent ? C3x (3x -1 x+2, 4x) a. C4, 2x -3) (ex) C2x) e [3 C3x-1) [2 d.) C3x-1) - 3e
1. Determine if these vectors are linearly independent or linearly dependent. If they are linearly independent show this. If they are linearly dependent, write a nontrivial linear combination (at least some coefficients need to be nonzero) of them being equal to 0. {4.0-6)}
please answer this
2 5 Let v V3 and w 2 -10 Determine if w is a linear combination of (v1, v2, v3) and, if so, find scalars (C1, C2, C3) such that w = c¡v1 + c2V2+ c3v3.
Plz solve both