Final answer:
To find a basis for the vector space of all 3 x 3 diagonal matrices, we can consider matrices of the form a11I + a22I + a33I, where I represents the identity matrix. Any scalar multiple of the identity matrix is linearly independent. Therefore, a basis for the vector space is {I, I, I}.
Step-by-step explanation:
To find a basis for the vector space of all 3 x 3 diagonal matrices, we need to determine which matrices satisfy the conditions of being diagonal and of size 3 x 3. A diagonal matrix has zero entries outside of its main diagonal. So, any matrix of the form:
A = a11I + a22I + a33I
where I represents the identity matrix, will be a 3 x 3 diagonal matrix. Since the identity matrix is linearly independent, we can conclude that any scalar multiple of the identity matrix will also be linearly independent. Therefore, a basis for the vector space of all 3 x 3 diagonal matrices is:
B = {I, I, I}