3.Let1 2-2 263T=3 451-1 1Determine whether T is linearly independent in the vector space M2,2(R) of all 2 x 2 matriceswith coefficients in R.

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Answered: 3. Let {[ 1 2 -2 2 6 3 T = 3 4 5 1 -1 1…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V = M² X2(R) be the vector space of all real 2 x 2 matrices. Determine if the following matrices are linearly independent; if linearly dependent, find a nontrivial linear combination of the form a₁A₁ + a₂A₂ + ª3A3 + ªА that results in the zero matrix. (Give a ₁, 2, 3, and a as real numbers. If the matrices are linearly independent, enter INDEPENDENT.) 1 (a₁₁²₂₁ A₁ = 10 00 A₂ = 01 00 0 =[:] 10 ₁ A3 = 3 4-[18] = 40 and A4
Plz answer all parts. Or leave it hanging
(a) The vector space M5×25(R) of 5 × 25 matrices contains a set of 150 linearly independent vectors. Answer (T/F): Justification:
Let 1 3 1 1 A = 3 9 15 9 1 3 9 5 (a) Find a basis for the column space of A. Answer: { Enter your answer as a vector or a list of vectors in parentheses separated by commas. For example (1,2,3,4),(5,6,7,8) (b) What is the dimension of the row space of A? (c) What is the dimension of the solution space of A? 4 -3 (d) Let v = a where a E R. Find the value of a such that v is in the solution space of A. 3 2 a = 0 NWO 3 2
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3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them
3. Find an example of a matrix that has Why can't this be a basis for the row space and the mullspace? 1. Explain why the vector v = H and as a basis for its row space and its column space. cannot be a row of A and also in the nullspace of A.
16. Show that if the non-zero vectors s), s2),..., s) are conjugate to a positive definite matrix G, then the vectors are independent.
7-8. Determine whether the space of the matrix A 2- I 2 O > Am 4 vector ū belongs A₁ = 0 to null
Do both

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3. Let {[ 1 2 -2 2 6 3 T = 3 4 5 1 -1 1 Determine whether T is linearly independent in the vector space M2.2(R) of all 2 × 2 matrices with coefficients in R.