Question

V is a vector space. Mark each statement True or False. Justify each answer.

a. R^2 is a two-dimensional subspace of R^3.
b. The number of variables in the equation Ax D 0 equals the dimension of Nul A.
c. A vector space is infinite-dimensional if it is spanned by an infinite set.
d. If dimV D n and if S spans V, then S is a basis of V.
e. The only three-dimensional subspace of R^3 is R^3 itself.

Answer 1

• False
• False
• False
• False
• True

Explanation:

A) False ; R^2 is not a two dimensional subspace of R^3 because Vector R^2 will have 2 entries while vector R^3 has 3 entries

B) False : The number of variable in the equation is not equal to the Dimension because dim of Null A represents the number of free variables

C) False : This is because you can span R^3 with an infinite set of vectors but that doesn't make it infinite dimensional subspace

D) False : This is only possible when S has exactly n elements/vectors

E) True : Since R^3 is a three dimensional subspace it is spanned with 3 vectors whom are linearly independent