part (6) 6. We are going to find a basis of all the polynomials in P4 that have 3 ,5 and 7 as a root. How?(1) Define a function T : P4 → R³ by setting[p(3)T(p(x)) = |p(5)[P(7)(2) Prove T is linear.(3) Find MT, the matrix of the transformation T.(4) Find a basis of Null(MT).(5) Use your previous answer to find a basis of Ker(t).(6) Explain why the basis of Ker(T) answers the question.

Answered: Image /qna-images/answer/244325ef-eed4-4a72-acc8-b627ce547d9d.jpg

Answered: (6) Explain why the basis of Ker(T)…

Math

Advanced Math

(6) Explain why the basis of Ker(T) answers the question.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 6CM: Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a...

See similar textbooks

Related questions

Q: -11 is a vector in the standard basis of R². 9.

Q: What is the basis and dimension of M3(R)?

A: Basis: For a vector space V, the set of linearly independent vectors that generate the whole vector…

Q: (d) Is {v1, 02, T3, 74} a basis for R4? Why?

Q: 3 Find the coordinate vector of a with respect to the basis B for R3. 2 1.

A: We find the coordinate vector.

Q: O Given Basis tunchiono Xn- X and scalar prodnct <ti9)= foga) da ure Gram- Schmi dt precedue t find…

A: The given sequence of functions are: xn=xn The first four functions are: x1=x1, x2=x2, x3=x3 and…

Q: Explain why S i not a basis for P2. S = {1, 4x, -5 + x?, 3x} O sis linearly dependent. O s does not…

A: We have to find out

Q: Find the coordinate vector of x2 + x - 17 relative to the basis {x² + 1, x + 2, x - 3}.

Q: he basic eigenvectors of A corresponding to the -1 0 1 -2 6 -3 3 -6 2 = -1 0 12 -18 | 0 0 6 -9…

Q: Verify that the polynomials 2 – x², x' – x, 2 – 3x² and 3 – x' form a basis for P3(x). Express cach…

Q: 1) Explain why 2 x 2 rotation matrices cannot be diagonalized (using real matrices). Your answer…

Q: Consider the quadratic form given by Q(r, y) = 8x2 + 6xy. 1. Give the symmetric matrix associated…

A: As per Bartleby's expert answering policy, we should answer only one question, so kindly post other…

Q: Find a basis fo

A: Given, A=3-916-26-161-38

Q: Find the coordinate vector of p(x) = 3 - x + 4x2 with respect to the basis B = {1, 1 + x, -1 + x2}…

Q: Find the change of basis matrix from B1 to B2. -1 2

Q: Find the stable vector of P =

A: Find the stable vector.

Q: Let V = Poly4(R) and let B = (1, x2 − 1, x, x^3 − 2x, x^4 − 3x^2 + 1). (a) Find a basis C of V such…

A: Let V = Poly4(R) and let B = {1, x2 − 1, x, x3 − 2x, x4 − 3x2 + 1}. Note that here V is a vector…

Q: Let TilP > Pz, the basis of NIT) T (ax²+bx+c) = 2Qx+ b. Then %3D is

A: Applying nullity definition we have

Q: Consider the polynomials p₁ (t)=3-t³, p₂ (t)=2-t², p₂ (t)=t-5t², and p₁ (t)=7-t+3t² −1³. 3 Is {P₁,…

Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

Q: Find an prthonormal basis of the plane x1+ 2x2- x3 0.. W

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: alative to Find He coordinate matix of standard basis the for M3,1. X =

Q: now that B = {x, r³ + 2x² + 4, x² + 2, 1} is a basis for P3.

Q: Using the principle of least-squares, Fit an equation of the form y = In(L(A,x)). where L(A, x) =…

Q: Find a basis of the subspace of R defined by the equation -7x1 – 5x2 – 4x3 = 0.

Q: a) Find the basis for the solution space of I1 +a2 - r3 = 0, -2r1 - 12 + 5r3 = 0, -11 + 4r3 = 0 %3D…

A: (a) given system of linear equation x1+x2-x3=0-2x1-x2+5x3=0-x1+4x3=0 to find - basis for solution…

Q: Expand x = (a, b, c, d) with respect to the orthogonal basis of: 2 -1 1 Vị = V2 -2 3 Express your…

A: Note: Observe that the given orthogonal basis spans only two dimension subspace of ℝ4, therefore the…

Q: Good morning, could you help me with that? Thank you very much Establish a vector basis for the set…

Q: y" +3y+Ay=0, y'(0) =0, y'(x)=0

Q: 1 A = 2 1 1 3 Find an orthonormal basis of the image of A.

A: A=-11-211-3 Orthonormal basis of A using the Gram Schmidt process. v1=-1-21, v2=11-3…

Q: Verify that {(sin θ, cos θ), (cos θ, −sin θ)} is an orthonormal basis for R2.

Q: c) Show that eigenvectors of [T]B exhibit an orthonormal basis of Rt.

Q: 1. Arguing geometrically, find an eigenbsis for the reflection about the line in R2 that is parallel…

A: The given equation formed for the basis is, 1-2 is, y=-2x The reflection for the equation y=-2x is,…

Q: Find an orthonormal basis of the plane x1 - 6x2 - x3 = 0. { }

Q: The vectors are an orthogonal basis for R2. -1 True False

A: Answer is False.

Q: Find

Q: 27. Find the dimension and one basis of the nullspace V = {(x1; x2; X3; T4) E R4 : x1 + x2 – x3 = 0…

A: As per our guideline we are supposed to answer only one question if multiple questions are posted.…

Q: 4. a) Give an example of a set of nonzero vectors in R3 that is linearly independent but is not a…

A: This question is related to Linear Algebra

Q: R4 Span {v1,..., V4}. Explain why {v1,..., V4} Is a basis for 4

Q: - --- Let B = Is Ba basis ofR ? Justify and x = %3D 2 3 3 your answer. If yes, compute xR [x]

Q: Find eijen values and eigen vectors 5 For M = 5 QNO:7 7

A: We need to find eigenvalues and eigenvectors for , M = 5775 For finding eigenvalues we need to find…

Q: Find a basis for the kernel of T.

A: In this question, we have a linear transformation by using the given linear transformation, we have…

Q: How is the basis {lk} picked?

Q: Prove that B is a basis of V.

Q: Show that the condition number of the square matrix, X (generated with X;j = 461-2), grows i+6j–2 as…

A: We will use the conventional definition of condition number of a matrix.

Q: 1. Find the vector x determined by the given coordinate vector [x]g and the given basis В. 8 B =…

Q: (a) Find the coordinate vector of x = - with respect to the ordered basis -4

Q: How would you represent it in Basis 2?

Q: After applying the Gram-Schmidt Orthonormalization Process to the basis B a 1 we get the basis B'…

Q: Find an orthonormal basis of the plane z + 7r2 - r3 = 0.

Concept explainers

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

part (6)

6. We are going to find a basis of all the polynomials in P4 that have 3 ,5 and 7 as a root. How?
(1) Define a function T : P4 → R³ by setting
[p(3)
T(p(x)) = |p(5)
[P(7)
(2) Prove T is linear.
(3) Find MT, the matrix of the transformation T.
(4) Find a basis of Null(MT).
(5) Use your previous answer to find a basis of Ker(t).
(6) Explain why the basis of Ker(T) answers the question.

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

Some notes.

VIEW

Solution.

VIEW

Step by step

Solved in 2 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Cartesian Coordinates$

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

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
Find the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: P2--->R2 defined by T(p(x)) B={x2,x,1},C , v = p(x) =a + bx + cx2
For the linear transformation R[x]≤ 3 ---> R[x]≤ 3 defined by T(p(x))=p(x-1), Find the matrix [T]B,B of T with respect to the basis: B: = (1, x, x2, x3)
Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is rotation by 5/4π. A = ?
How can I find [L] from basis B to basis C, if basis B = {1 + x + x^2, 1 + x, 2x + x^2} and basis C = {1 + x, x - x^2, 3 + x^2}? L(p) = p + p', meaning that the linear transformation is the up to 2nd degree polynomial plus its own derivative. Thank you in advance.
1. Find the matrix A of the linear transformation T from R2 to R2 that rotates any vector through an angle of 45∘ in the counterclockwise direction. 2. To every linear transformation T from R2 to R2, there is an associated 2×2 matrix. Match the following linear transformations with their associated matrix.
Suppose V is a vector space of finite dimension k and T : V → V is a linear transformation. Suppose T has one single eigenvalue λER, and call n its geometric multiplicity. Which value(s) of n (if any) will result in [T]% being similar to a diagonal matrix, for all bases a of V? Your "values" of n should be in terms of k. If no such value of n exists, explain why not. Clarification 1: In general there are two meanings of multiplicity that apply to eigenvalues. "Multiplicity of an eigenvalue" could mean "number of times (x - 2) appears in the characteristic polynomial x(x) ("Algebraic Multiplicity") or the dimension of the eigenspace E₁ ("Geometric Multiplicity"). In this question, we are considering the \textbf{geometric multiplicity} n of λ. Clarification 2: Saying "There exists basis a of V for which [T] is diagonal" is equivalent to saying that "[T] is similar to a diagonal matrix for all bases a of V". Hint: Your solution can consider k>n, k = n, and k < n separately.
Find a linear transformation T from R4 to R4 such that the kernel of T has dimension 2 and give a basis for the range of T. What is a basis for the kernel of T?
The linear transformation T : R2 → R2 with matrixA = cos θ − sin θsin θ cos θ rotates vectors in the x y-plane counterclockwise through an angle θ, where 0 ≤ θ <2π. By arguing geometrically, determine all valuesof θ for which A has real eigenvalues. Find the realeigenvalues and the corresponding eigenvectors.