Simple True or False Linear Alegbra 1 Suppose A and B are bo

Simple True or False (Linear Alegbra)

1) Suppose A and B are both subspaces of a vector space V. Then A B must be a subspace of V.

2) If B is finite subset of a vector space V and A is also a subset of V that contains B. Then it must be true that Span(B) is a subspace of Span(A).

3) A set that contains the zero vector can\'t be linearly independent.

4) If A is an nxn matrix then the set of all column vector X Rⁿ such that AX=0 is a subspace of Rⁿ.

5) Let A be the set of all vectors of the form (v_1,v_2, 0) R³. Then A is a subspace of R³.

6) The set of all polynomials with real coefficients that have degree two is a subspace of P₂.

7) The set S={t(1,0,0)+(0,0,1): t is a real number} is a subspace of R³.

Solution

Solution:-

1)True,If A and B are both subspaces of a vector space V,then intersection of A and B is a vector space of V.

2)True, If b is a finite subset of a vector space V and A is a subset of V, which contains B, then Span(B) is a subspace of Span(A).

3)True.By the properties of vector space, a set that contains the zero vector cannot be linearly independent.

4)False,Ax=0 means, the vector is linearly independent,so it is not a subspace of Rn

5)True,as the set of all vectors which are parallel to given plane is a subspace

6)False,the set of polynomials of degree 2 is not a subspace of C[0,1].

7)True,the given set is a subspace of R3, as t is a real number.