Which of the following subsets of 33 are subspaces of 33 Whi

Which of the following subsets of ?3×3 are subspaces of ?3×3?

Which of the following subsets of R^3 times 3 are subspaces of R^3 times 3? The 3 times 3 matrices whose entries are all integers The 3 times 3 matrices whose entries are all greater than or equal to 0 The symmetric 3 times 3 matrices The 3 times 3 matrices with determinant 0 The 3 times 3 matrices with all zeros in the second row The diagonal 3 times 3 matrices

Solution

A. The set of all 3 x 3 matrices , all whose entries are integers is closed under addition, but not under scalar multiplication, when the scalar is non-integer real number ( e.g. a rational number). Therefore, this set is not a vector space.

B. The set of all 3 x 3 matrices , all whose entries are 0, is closed under addition, but not under scalar multiplication, when the scalar is a negative real number or a rational number less than 1. Therefore, this set is not a vector space.

C. The set of all 3 x 3 symmetric matrices is a vector space as the set is closed under both addition and scalar multiplication.

D. The set of all 3 x 3 matrices whose determinant is 0 is not closed under addition. Therefore, the set is not a vector space. ( e.g. when we add 2 matrices A and B with rows A: 1,0,0 ; 2,0,1 ; 3,0,2 ; B: 0,1,0 ; 2,0,0 ; 3,0,0 , the resuting matrix A + B : 1,1,0 ; 4,0,1; 6, 0,2. Now. det A = det B = 0, but det ( A + B) = -2 0.)

E. The set of all 3 x 3 matrices whose second row is all zeros is a vector space as it is closed under both addition and scalar multiplication.

F. A diagonal matrix is a matrix (usually a square matrix) in which the entries outside the main diagonal () are all zero. Thus, the sum of two 3 x 3 diagonal matrices is a 3 x 3 diagonal matrix. Also, a scalar multiple of a 3 x 3 diagonal matrix is a 3 x 3 diagonal matrix. Therefore, The set of all 3 x 3 diagonal matrices , being closed under addition and scalar multiplication, is a vector space.