It is a mapping, a function,įrom Rn to Rm. Redundant with the closure under scalar multiplication. Will always write, oh and the zero vector has toīe a member of V. Vectors is also in V, I could just set the scalar And why I say that's redundant,īecause if I say that any multiple of these With n components here, because V is a subspace of Rn. Statement is that V, well it must contain the zero vector. And we sometimes call thisĬlosure under scalar multiplication. This is also going to be a member of our subspace. Of them, let's say a, and I multiply a by some scalar, that Of our subspace - we also know that if I pick one
Subspace by a scalar - so the fact that those guys are members Subspace, we also know that if we multiply any member of our
#Vector subspace definition plus
Vectors, or a plus b, is also in my subspace. Subspace, we then know that the addition of these two So let say I take the members a and b- they're both Subset of Rn where if I take any two members of that subset. What does it mean? That's just some set, or some In fact, I'm not even sure why Sal lists it as a rule of subspaces that they include the 0 vector because with rule #2 or rule #3 they have to include the zero vector anyways. Well, if A and -A are both in our subspace, then so must A+ (-A)… which is of course, the zero vector. After all, if A is a vector in our subspace, and so is -1*A (from rule #2) then the subspace must also include a zero vector because if vector addition holds, then the sum of any two vectors in our subspace must ALSO be in our subspace. In other words, rule #1 must hold if rule #2 is to hold.Īnd honestly, rule #1 also must hold if rule #3 is to hold. So for rule #2 to hold, the subspace must include the 0 vector. Well suppose we multiply by the scalar 0? We would get the 0 vector. If rule #2 holds, then the 0 vector must be in your subspace, because if the subspace is closed under scalar multiplication that means that vector A multiplied by ANY scalar must also be in the subspace. Well, imagine a vector A that is in your subspace, and is NOT equal to zero. Remember the three rules that Sal gave for the definition of a subspace? They were:
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