Question

Let V be the set of all 3x3 matrices with Real number entries, with the usual definitions of scalar multiplication and vector addition. Consider whether V is a vector space over C. Mark all true statements (there may be more than one).

a. The scalar closure axiom is satisfied
b. The additive inverse axiom is not satisfied
c The additive inverse axiom is satisfied
d. The additive closure axiom is not satisfied
e. The scalar closure axiom is not satisfied
f. The additive closure axiom is satsified
g. V is not a vector space over C
h. V is a vector space over C
i. The zero axiom is satisfied
j. The zero axiom is not satisfied

Answer 1

• the Scalar Closure axiom is not satisfied

• V is not a Vector Space of C

• The Additive Closure axiom is satisfied.

Explanation:

According to the Question,

• Given That, Let V be the set of all 3x3 matrices with Real number entries, with the usual definitions of scalar multiplication and vector addition. Consider whether V is a vector space over C.

• For V is a vector space over C and V is Set of 3x3 Matrices with Real entries.

Then, For any u,w ∈ V ⇒ u+w ∈ V

And u∈V and z∈C ⇒ z u ∈ V

So, If we take any z= i ∈ C

• and V be any 3x3 matrices with Real entrices.

then, z,v ∉ V ∴z,v Has Complex entries

So, the Scalar Closure axiom is not satisfied

Hence, V is not a Vector Space of C

Any u,w ∈ W ⇒ u+w ∈ V

So, The Additive Closure axiom is satisfied.