Question

Can someone please explain me the difference between vector spaces and subspaces?

Answer 1

there is no difference. subspaces are vector spaces.

a subspace is a vector space within a "bigger" vector space.

Answer 2

Well, to add ... its like saying that the subspace S is contained within vector space V just like your drawing room is inside your home. They have a set/subset relationship... but better like saying that the dimensions may differ. (The 2D floor of your house is contained in your 3D home) ..

Answer 3

vector space definition is clear. subspace means that it is itself a vector space but it's part of larger space. Such as you can have a line (1D) lying inside a plane (2D). Naturally, there are many direction for such a line. It is usually more useful to think visually than from algebraic point of view.

Answer 4

In simple words,Sub space is nothing but a part of vector space with it's own characteristics,like usual addition and scalar multiplication.and also satisfy the conditions of original vector space.
Ex. \[W={ (a,0,0)/a \in R} , V=R^{3}\]
in above example W be the subspace of V.
In other exampl

Answer 5

One of the crucial characteristics of a subspace is that it contains every combination of any vector within it -- so if v and w are in subspace S, then v+w and any cv or cw is also in subspace s. That means it must contain the zero vector, since 0*v and 0*w must also be in S, and v-v and w-w must also be in S.

So, not just any line is a subspace - but a line that passes through 0.

Answer 6

See page 122 in the textbook.