1. Use the definition of a subspace to prove or disprove: W is a subspace of M, (you must verify all 4 axioms if it's a subspace) [a+4b 0 W is the set of all 3x2 matrices of the form a 3bl

Let W be the subspace of M₁0.10 (i.e., 10 × 10 matrices) consisting of all matrices whose diagonal entries are zero. Find the dimension of W.

Let P₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 14x² − 17x + 10, 17x – 19x² – 11 and 9x² - 10x + 6. a. The dimension of the subspace H is b. Is {14x² - 17x + 10, 17x - 19x² – 11,9x² – 10x +6} a basis for P₂? choose c. A basis for the subspace H is { (where you can enter xx in place of x²) Be sure you can explain and justify your answer. }. Enter a polynomial or a comma separated list of polynomials

s) The trace of a square n x n matrix A = (a) is the sum a₁ + a22+...+ aan of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0. Is H a subspace of the vector space V? 1. Is H nonempty? choose 12 5 6 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that H is not closed 3 4 7 under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) " 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in IR and a matrix in H whose product is not in H 3 4 using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that is not closed 56 under scalar multiplication, it is…