Mark each statement True or False. Justify each answer. (a) A vector is any element of a vector space. (b) If
is a vector in a vector space V, then
is the same as the negative of
. (c) A vector space is also a subspace. (d)
is a subspace of
. (e) A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii)
,
, and
are in H, and (iii) c is a scalar and
is in H.
The Correct Answer and Explanation is:
Here is the correct evaluation of each statement with justification:
(a) A vector is any element of a vector space.
True
A vector space is defined as a set of vectors along with operations of addition and scalar multiplication that satisfy certain properties like associativity, distributivity, and the existence of a zero vector. By definition, any element belonging to the set of a vector space is called a vector. It is important to note that in mathematics, the term “vector” does not only refer to geometric arrows but applies to any element within a vector space, whether they are real numbers, functions, or matrices.
(b) If 𝑣 is a vector in a vector space V, then 𝑣 is the same as the negative of 𝑣.
False
For any vector 𝑣 in a vector space V, the negative of 𝑣 is denoted by -𝑣. This is defined as the unique vector that satisfies the equation 𝑣 + (-𝑣) = 0, where 0 is the zero vector of the space. Therefore, unless 𝑣 is the zero vector itself, 𝑣 is not the same as -𝑣. For example, if 𝑣 is a nonzero vector, 𝑣 and -𝑣 point in opposite directions in geometric terms.
(c) A vector space is also a subspace.
True
A subspace is defined as a subset of a vector space that is itself a vector space under the same operations. The entire vector space V is considered a subspace of itself because it trivially satisfies all the conditions required of a subspace, such as closure under addition and scalar multiplication, containing the zero vector, and so on.
(d) {0} is a subspace of V.
True
The set containing only the zero vector, {0}, is always a subspace of any vector space V. It contains the zero vector by definition, is closed under addition because 0 + 0 = 0, and is closed under scalar multiplication because c * 0 = 0 for any scalar c. This set is known as the trivial subspace.
(e) A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) 𝑢, 𝑣, and 𝑢 + 𝑣 are in H, and (iii) c is a scalar and c𝑢 is in H.
False
The conditions provided are incomplete or miswritten. The correct conditions for a subset H to be a subspace of V are:
- The zero vector of V is in H.
- H is closed under addition, meaning for all 𝑢, 𝑣 in H, the sum 𝑢 + 𝑣 is also in H.
- H is closed under scalar multiplication, meaning for all 𝑢 in H and any scalar c, the product c𝑢 is in H.
The given wording incorrectly mentions 𝑢, 𝑣, and 𝑢 + 𝑣 being in H rather than stating closure under addition for any two elements. Therefore, the statement as written is false.
