Which of the following subsets of RXSare subspaces of RXA. The 3 x 3 matrices of rank 2B. The diagonal 3 x 3 matrices8.C. The 3 x 3 matrices A such that the vector 7 is in the kernel of AD. The 3 x 3 matrices whose entries are all greater than or equal to 0E. The 3 x 3 matrices with all zeros in the second rowF. The 3 x 3 matrices whose entries are all integers

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Answered: Which of the following subsets of…

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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Which of the following subsets of R³X3 are subspaces of R³x3? A. The diagonal 3 x 3 matrices B. The 3 x 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries) 5 (-6) 2 C. The 3 x 3 matrices A such that A D. The 3 x 3 matrices in reduced row-echelon form E. The 3 x 3 matrices whose entries are all greater than or equal to 0 F. The 3 x 3 matrices of determinant 2
s) The trace of a square n x n matrix A = (a) is the sum a₁ + a22+...+ aan of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0. Is H a subspace of the vector space V? 1. Is H nonempty? choose 12 5 6 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that H is not closed 3 4 7 under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) " 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in IR and a matrix in H whose product is not in H 3 4 using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that is not closed 56 under scalar multiplication, it is…
Let W be the subspace of all diagonal matrices in M2.2. Find a basis for W. Then give the dimension of W. If you need to enter a matrix as part of your answer, write each row as a vector. For example, write the matrix ab c d as {(a, b), (c, d)}
[-3 6. Let B = -1 1 1 -2 2 3 -11. Determine a basis for, and dimension of, the following -4 5 subspaces: a. The column space B. b. The row space of B. С. The null space of B.
A square matrix A is nilpotent if A" : = 0 for some positive integer n. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer [1 2] [56] 3.6 . (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A 3 4 and B such that (A + B)" ‡ 0 for all positive integers n.) 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer [34] 2, 3] (Hint: to show that is not closed under scalar multiplication, it is sufficient to find a real…
✓ *Which of the following subsets of R³x³ A. The 3 x 3 matrices A such that the vector are subspaces of R³×³? (-) 8 is in the kernel of A 5 B. The 3 x 3 matrices with all zeros in the second row C. The 3 x 3 matrices whose entries are all greater than or equal to 0 D. The 3 x 3 matrices with determinant 0 E. The 3 x 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries OF. The x 3 matrices whose entries are all integers
Determine which of the following sets are subspaces of R². 1. {[1, ₂] | 1 = -922}. 2. {[1, 2] | 6x₁ + 3x₂ = 0}. 3. {[1, ₂] | ₁ + x₂ > 0}. Determine which of the following sets are subspaces of R²x2. ? ? ? ? ? ? V 1. The 2 x 2 matrices with determinant equal to zero. ✓2. The 2 x 2 matrices whose entries are all greater than or equal to zero. ✓3. The 2 x 2 lower triangular matrices. Note: You can earn partial credit on this problem.

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