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4. Consider the vector space M of 2 x 3 matrices. (a) Define the subset SC M S = -{(a = a b = c) €M | a,b,c,dER} a 0 b-c d-a 0 d-c i. Show that S is closed under addition ii. Show that S is closed under scalar multiplication iii. Determine if S is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace. (b) Let P be a fixed (but unknown) 3 x 2 matrix. Define a new subset Tp C M Tp = {A E MAP = PA} i. Show that Tp is closed under addition ii. Show that Tp is closed under scalar multiplication iii. Determine if Tp is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace.

4. Consider the vector space M of 2 x 3 matrices. (a) Define the subset SC M S = -{(a = a b = c) €M | a,b,c,dER} a 0 b-c d-a 0 d-c i. Show that S is closed under addition ii. Show that S is closed under scalar multiplication iii. Determine if S is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace. (b) Let P be a fixed (but unknown) 3 x 2 matrix. Define a new subset Tp C M Tp = {A E MAP = PA} i. Show that Tp is closed under addition ii. Show that Tp is closed under scalar multiplication iii. Determine if Tp is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace.

Advanced Engineering Mathematics
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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4. Consider the vector space M of 2 x 3 matrices. (a) Define the subset SCM S b-a 0 b-c = { (a = a b=c) €M | a,b,c,d€R} € M | a, b, c, d €R d-a 0 d-c i. Show that S is closed under addition ii. Show that S is closed under scalar multiplication iii. Determine if S is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace. (b) Let P be a fixed (but unknown) 3 x 2 matrix. Define a new subset Tp M Tp = {A € M | AP = PA} i. Show that Tp is closed under addition ii. Show that Tp is closed under scalar multiplication iii. Determine if Tp is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace.
Transcribed Image Text:4. Consider the vector space M of 2 x 3 matrices. (a) Define the subset SCM S b-a 0 b-c = { (a = a b=c) €M | a,b,c,d€R} € M | a, b, c, d €R d-a 0 d-c i. Show that S is closed under addition ii. Show that S is closed under scalar multiplication iii. Determine if S is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace. (b) Let P be a fixed (but unknown) 3 x 2 matrix. Define a new subset Tp M Tp = {A € M | AP = PA} i. Show that Tp is closed under addition ii. Show that Tp is closed under scalar multiplication iii. Determine if Tp is a subspace of M. If so, find a basis and the dimension of the subspace. If not, provide a reason why it fails to be a subspace.
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