Linear Algebra True of False not necessary to prove but if y

Linear Algebra: True of False (not necessary to prove, but if you want to that\'s fine)

Solution

a) A being a mXn vector and non zero vector b such that R^n

the resultant vector will be mXn * (nX1) hence b must be a vector of mX1, hence it is not necessary that it will be sub-space of R^(n)

It can also be possible that for values of m>n, then in that case it will be a higher space than R^n

Hence first statement is False

b) The second statement is True since we can write v vector as

v = aw1 + bw2 + cw3 + .... + kkkwm

since the vector v is linear combination of all w vector

Hence v must be spanned by all w1, ..., wm

c) Since there are three vectors, and hence if they all are linearly independent then they can scan the complete surface R^(3)

in that case, span{v1,v2,v3} will be complete R^(3) and not the sub-space of lower dimension

Hence third statement is False