FOR EXTRA CREDIT # 1: Determine whether the following subset of the vector space

Question

FOR EXTRA CREDIT # 1: Determine whether the following subset of the vector space Mn,n (the set of all n x n matrices) is a SUBSPACE of Mn,n, with the standard operations of matrix addition and scalar multiplication The subset W of all n x n matrices A that commute with a given matrix B, i.e., AB-BA You need to show that W satisfies the three properties defining a subspace, SO FOLLOW THESE STEPS STEP 1. Show that W is not empty: Show that the zero matrix O is in W by showing that OB-BO, where OB and BO both equal the same matrix according to how the zero matrix works when it is multiplied by another matrix. show in steps why using Theorem 2.3, p. 54, Section 2.2, text. matrix A and a scalar c e R, show in steps why STEP 2. Show that W holds under closure under addition: For arbitrary n × n matrices A and C, (A+C)2 = .. . = B(A + C) STEP 3. Show that W holds under closure under scalar multiplication: For an arbitrary n x n using Theorem 2.3, p. 54, Section 2.2, text. HINT: See the Solutions to HW # 9 on Section 4.3.]

Explanation / Answer

W is the subspace of Mn,n matrix which commute with a given matrix B.

We have O (null matrix of order n,n) and we know that OB =O;

Also BO=O;

So we have OB=BO;

=> O Belongs to W.

=> W is non empty.

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Now let A and C belongs to W then we have

(A+C)B=AB+CB [matrix addition is distributive]

= BA+BC [definition of W]

=B(A+C) [distribution law in matrix]

=> (A+C) belongs to W.

______________

c be any scaler then we have

For A belongs to W

(cA)B=c(AB) [associative law]

=c(BA) [definition of W]

=(cB)A [associative]

=(Bc)A

=B(cA)

=> cA belongs to W.

So, W is a subspace.

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