Question

Every nonzero vector space V contains a nonzero proper subspace
true
false

Answer 1

please explain your ans

Answer 2

false

Answer 3

how false

Answer 4

look at the vector space that has the basis vector \{(1,0)\}


so V=span( \{(1,0)\})

the only proper subspace of V is the zero subspace

Answer 5

\[V=\text{span}(\{(1,0)\})\]

Answer 6

do you see why?

Answer 7

plz explain if u can ''the only proper subspace of V is the zero subspace''

Answer 8

Suppose W is a proper subspace of V and that W is not the zero subspace....

Answer 9

then there is a nonzero vector \(w\in W\)

Answer 10

the vector \(w\) has the form (x,0) where \(x\ne 0\)

then \[(x,0)/x=(1,0)\in W\]


thus W=V

...a contradiction

Answer 11

what is proper subspace? is it like proper subset

Answer 12

sort of

Answer 13

W is a proper subspace of V if W is a vector space and every vector in W is in V, but W\(ne\)V

Answer 14

W\(\ne\)V

Answer 15

means proper subspace is actually a proper subset in which properties of vector space is satistied right?

Answer 16

yes

Answer 17

ok thanks a lot

Answer 18

np

Answer 19

@zarkon please help if u can.
Similar matrices represent the same linear transformation. please prove it or send me a link of its prove its urgent....

Answer 20

look at

http://mathprelims.wordpress.com/2009/06/25/definition-of-similarity-using-linear-transformations/