3. Let A be an arbitrary m x n matrix. Consider all x's, such that Ax = 0, and show that the set of all these x's is a subspace of R". Please show your work here and DO NOT simply state a theorem.

Let A be a fixed 10 x 10 matrix and let H be the set of all 10 x 10 matrices N with the property that NT A = A N. That is, H = {N | NT A = AN}. Is H a subspace of all 10 x 10 matrices? Justify your answer. Reminder: (A + B) = A + BT and (CA) = c (AT)

Find the orthogonal projection of x³ € P3 onto the subspace Span (1, 1 + x, 1 + x²). Justify your answer!

Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector orthogonal to W. 3 14-04 y = U₁ -2 y = (Type an integer or simplified fraction for each matrix element.)

I need to prove that this statement is fals but I don't know how.

Let A be any 5x7 matrix for which the col(A) has dimension 3, calculate: the nullity(A), and, state which vector space R^k that null(A) is a subspace of (give k). O A. nullity(A)=2, k=7 B. nullity(A)=4, k=7 OC. nullity(A)=2, k=5 O D. nullity(A)=4, k=5

Let A be any 5x7 matrix for which the​ col(A) has dimension​ 3, calculate: the​ nullity(A), ​and, state which vector space​ R^k that​ null(A) is a subspace of​ (give k).   A. ​nullity(A)=4, k=7   B. ​nullity(A)=2, k=5   C. ​nullity(A)=4, k=5   D. ​nullity(A)=2, k=7