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Let event A be the coin landing on heads and let event B be rolling a 5 on a six-sided die.
Events A and B are independent if, and only if:
It is given in the question that the above condition for independence is met.
Also A and B are independent if:
P(A|B) = P(A)
P(A) = 1/2
Therefore the probability of flipping a coin and it landing on heads, given that you rolled a 5 on a six-sided die is 1/2. The two events are independent.
Events A and B are independent if, and only if:
It is given in the question that the above condition for independence is met.
Also A and B are independent if:
P(A|B) = P(A)
P(A) = 1/2
Therefore the probability of flipping a coin and it landing on heads, given that you rolled a 5 on a six-sided die is 1/2. The two events are independent.
A vector space is a subspace of a vector space
if
is non-empty,
- for any two vectors
we have
, and
- for any scalar
and
we have
.
It's easy to show the first condition is met by all the sets in parts (a-g).
(a) is a subspace of because adding any 2x2 diagonal matrices together, or multiplying one by some scalar, gives another diagonal matrix.
(b) and (c) are also subspaces for the same reasons.
(d) is not a subspace because because this set of matrices does not contain the zero matrix.
(e), however, is a subspace. Any linear combination of matrices in this set always yields a matrix with 0 in row 1, column 1 entry.
(f) is a subspace. A symmetric matrix is one of the form
Adding two symmetric matrices gives another symmetric matrix:
(g) is not a subspace. Consider the matrices
Both matrices have determinant 0, but their sum is the identity matrix with determinant 1.