Let W be a subset of Rn What does it mean when we say that W

Let W be a subset of R^n. What does it mean when we say that W is closed under scalar multiplication? Whenever X is in R^n and c is a scalar, then cX in R^n. Whenever X is in R^n and c is a scalar, then cX in W. If cX in W and X is in W, then c is a scalar. Whenever X is in W and c is a scalar, then cX is in W. Whenever X is in W and c is a scalar, then cX is in R^n. Let S be a subset of R^n. What does it mean when we say that S is linearly independent? S is a basis. S is closed under both addition and scalar multiplication. Every vector in R^n is a linear combination of vectors in S. The only way to write O as a linear combination vectors of S is the zero combination (where one takes zero multiples of each vector of S).

Solution

1)

Option c is not fully correct as if cx is in W and c is a scalar, then x is in W.

Being closed under scalar multiplication means that vectors in a vector space, when multiplied by a scalar (any real number), it still belongs to the same vector space.

So, if X is in W and W is a suset of R^n then cX would alos be in the vector space of W

Option d)

2) Option c and d may be true but :

Option d