TRUE or FALSE LINEAR ALGEBRA a A vector space must contain

TRUE or FALSE ( LINEAR ALGEBRA)

a) A vector space must contain at least two vectors.

b)The set of all 3 × 2 matrices with the standard matrix addition and scalar multiplication is a vector space.

c) Every subspace of a vector space is a vector space.

d) Every vector space is a subspace of itself.

e) The intersection of any two subspaces of a vector space is a subspace.

f) The union of any two subspaces of a vector space is a subspace.

g) The span of any two vectors in a vector space is a subspace.

h) Two sets of vectors that span the same subspace must be equal.

i) The solution set of a homogeneous linear system of equations of n unknowns is a subspace of Rⁿ.

j) A line through the origin is a subspace of Rⁿ.

Solution

Post multiple questions to get the remaining answers

a) The first statement is FALSE

since the vector space can also contain only one vector i.e. the zero vector, hence it is not necessary for the vector space to have atleast two vectors

b) The second statemnt is TRUE

Let us define the below matrix as 3X2 matrix

M1 + M2

=>

which will belong to the set of 3X2 matrix, similarly we can prove that K*M1 will belong to the 3X2 space where K is a constant

It will belong to 3X2 matrix with the elements multiplied by a constant factor

c) The third statement is also true, since the subspace of a vector space is a vector space