State whether each of the following statements is always tru

State whether each of the following statements is (always) true, or is (possibly) false, in the box after the statement.

• If you say the statement may be false, you must give an explicit example - with numbers, matrices, or functions, as is appropriate!

• If you say the statement is always true, you must give a clear explanation.

a) If {v₁, v₂, v₃} is linearly dependent in R³ , then {v1, v2} is also linearly dependent in R³.

b) The set of polynomial functions : {1 + ax² | a R} is a subspace of F(R) = {f | f : R R}.

c){ [ a b

c   d ] M_2,2(R) | a + d = 0} is a subspace of M_2,2(R).

d) if v₁,v₂, v₃, and v₄ are linearly independent vectors in a vector space V, then dim(V)=4

Solution