Let-4-1 017-113-1A13-11-313-1 1-3Find orthonormal bases of the kernel and image of A.Basis of the kernel:1-171{|| Basis of the image:-41-174

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Answered: Let -4 -1 7 -11 3 -1 A 13 -1 1 -3 13 -1…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 69E: Find a basis for R2 that includes the vector (2,2).

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Find a basis for R2 that includes the vector (2,2).
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Find : (1) a basis and the dimension of the column space of A . (2) a basis and dimension of the null space of A .
Let A be an n ×n matrix with complex entries. Prove that AA∗= I if and only if the rows of A form an orthonormal basis for Cn.
A)Show that B is a basis for P2 second-degree polynomial space B)Obtain the characteristic polynomial, the eigenvalues and eigenvectors of the following matrix C)Determine whether the following set of vectors is orthogonal. Then construct an orthonormal basis for R3 (THREE-DIMENSIONAL VECTOR)
6. Let (image):(a) Find a basis for the null space of A.(b) Find a basis for the column space of A.(c) What are the rank and nullity of A?(d) Show that the rank-nullity theorem is verified.
which is the collection of all polynomials of degree ≤ 3. Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2) Why is the following true? If {p1, p2, p3} spans P2 then it is a basis for P2. (1) Let p1 = 2−x+x2 , p2 = 1+x, p3 = x+x2 . Show that S = {p1, p2, p3} spans P2. Conclude that S is a basis for P2. (5) Using (2.3) or otherwise, write p = 3 + 5x − 4x2 as a linear combination of p1, p2 and p3. Show all working. Hence ﬁnd (p)S, the coordinate vector of p relative to S. (2) Explain why are the vectors q1 = 8 + 4x − 6x2 and q2 = −4 − 2x + 3x2 are linearly dependent in P2? (2)
Let R⁴ have a product in Euclidean. Use the Gram Schmidt process to convert the basis (u1, u2, u3, u4) to an orthonormal basis if u1= (0, 2, 1, 0) u2 = (1, -1, 0, 0), u3 = (1, 2, 0, -1), u4= (1, 0, 0, 1)
Find a basis for the kernel of AA
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Let S ={v1 , v2 ,...,vk } be an orthonormal basis for the Euclidean space V and {a1 ,a2 ,...,ak } be any set of scalars none of which is zero. Prove thatT ={a1 v1 ,a2 v2 ,...,ak vk }is an orthogonal basis for V . How should the scalars a1,a2 ,...,ak be chosen so that T is an orthonormal basis for V ?

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Let -4 -1 7 -11 3 -1 A 13 -1 1 -3 13 -1 1 -3 Find orthonormal bases of the kernel and image of A. Basis of the kernel: -1 -17 1