Determine whether the following are subspaces of R2×2 : (a) Thesetofall2×2diagonalmatrices (b) The set of all 2 × 2 triangular matrices (c) The set of all 2 × 2 lower triangular matrices (d) Thesetofall2×2matricesAsuchthat a12=1 (e) The set of all 2 × 2 matrices B such that b11 = 0 (f) The set of all symmetric 2 × 2 matrices (g) Thesetofallsingular2×2matrices

A vector space is a subspace of a vector space if

It's easy to show the first condition is met by all the sets in parts (a-g).

(a) is a subspace of because adding any 2x2 diagonal matrices together, or multiplying one by some scalar, gives another diagonal matrix.

(b) and (c) are also subspaces for the same reasons.

(d) is not a subspace because because this set of matrices does not contain the zero matrix.

(e), however, is a subspace. Any linear combination of matrices in this set always yields a matrix with 0 in row 1, column 1 entry.

(f) is a subspace. A symmetric matrix is one of the form

Adding two symmetric matrices gives another symmetric matrix:

(g) is not a subspace. Consider the matrices

Both matrices have determinant 0, but their sum is the identity matrix with determinant 1.

true

step-by-step explanation: