4.1 vector spaces and subspaces

Subspace of a vector space V

A subset H of V that has 3 properties: 1. The zero vector of V is in H 2. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in H 3. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H

If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V

False, it needs to be f(t) = 0 for all t

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, parts​ (ii) and​ (iii) should state that u and v represent all possible elements of H.

R2 is a subspace of R3

False, the elements in R2 aren't even in R3.

A subset H of a vector space V, is a subspace of V if the zero vector is in H

False, the set must also be closed under addition and scalar multiplication

A vector is an arrow in three-dimensional space

False, this is an example of a vector, but there are certainly vectors not of this form.

ℝ2 is a subspace of ℝ3.

False, ℝ2 is not even a subset of ℝ3.

Axioms for a vector space

In the following​ axioms, u​, v​, and w are in vector space V and c and d are scalars. 1. The sum u+v is in V. 2. u + v = v + u 3. ​(u + v​) + w = u + ​(v + w​) 4. V has a vector 0 such that u + 0 = u. 5. For each u in​ V, there is a vector −u in V such that u + (−u​)=0. 6. The scalar multiple cu is in V. 7. ​c(u + v​) = cu + cv 8. ​(c + ​d)u = cu + du 9. ​c(du​) = ​(cd)u 10. 1u = u

Determine if the given set is a subspace of ℙn. All polynomials of degree at most 3, with integers as coefficients

No, it is not closed under multiplication by scalars

Determine if the given set is a subspace of ℙn. The set of all polynomials of the form p​(t) = a + t^2​, where a is in ℝ.

No, not a subspace Pn for any n, it satisfies neither the 2nd nor 3rd condition given in the definition of a subspace

For fixed positive integers m and​ n, the set Mm×n of all m×n matrices is a vector​ space, under the usual operations of addition of matrices and multiplication by real scalars. Let F be a fixed 3×2 ​matrix, and let H be the set of all matrices A in M2×4 with the property that FA = 0 ​(the zero matrix in M3×4​). Determine if H is a subspace of M2×4.

The set H is a subspace of M2×4 because the set contains the 2×4 zero​ matrix, the set is closed under​ addition, and the set is closed under multiplication by scalars.

Determine if the given set is a subspace of ℙn. The set of all polynomials of the form p​(t) = at^2​, where a is in ℝ.

The set is a subspace of ℙ2. The set contains the zero vector of ℙ2​, the set is closed under vector​ addition, and the set is closed under multiplication by scalars. Recall the definition of a subspace. A subspace of a vector space V is a subset H of V that has the three following properties. a. The zero vector of V is in H. b. H is closed under vector addition. That​ is, for each u and v in​ H, the sum u + v is in H. c. H is closed under multiplication by scalars. That​ is, for each u in H and each scalar​ c, the vector cu is in H. Consider the set of all polynomials of the form p​(t) = at^2​, where a is in ℝ. The zero vector of ℙ2 occurs in this set when a = 0. The sum of two vectors in the​ set, rt^2 and st^2​, is ​(r + ​s)t^2. This is also in the set. The set is closed under vector addition.​ Finally, multiplying one vector in the​ set, kt^2​, by a​ scalar, m, yields mkt^2​, which is also in the set. The set is closed under multiplication by scalars.​ Therefore, the set of all polynomials of the form p​(t) = at^2​, where a is in ℝ​, is a subspace of ℙ2.

Determine if the given set is a subspace of ℙn. The set of all polynomials in ℙn such that p​(0) = 0

The set is a subspace of ℙn because the set contains the zero vector of ℙn​, the set is closed under vector addition, and the set is closed under multiplication by scalars.

If u is a vector in a vector space​ V, then ​(−​1)u is the same as the negative of u.

The statement is true. For each u in​ V, there is a vector −u in V such that −u = ​(−​1)u.

If u is a vector in a vector space V, then (-1)u is the same as the negative of u.

True

A vector space is also a subspace

True (Its always a subspace of itself, at the very least.)

A vector is any element of a vector space.

True by the definition of a vector space

A vector is any element of a vector space.

True, The elements of a vector space are called vectors.

A vector space is also a subspace of itself.

True, the axioms for a vector space include all the conditions for being a subspace.

A subspace is also a vector space

True, this is the definition of subspace, a subset that satisfies the vector space properties.

If u and v are in V, is u + v in V?

Yes, if u and v are in V, their entries are nonnegative. Since a sum of nonnegative numbers is nonnegative, the vector u + v has nonnegative entries. Thus u + v is in V

Determine if the given set is a subspace of ℙn. All polynomials in Pn such that p(0) = 0

Yes. The zero vector is in the set H. If p and q are in H, then (p + q)(0) = p(0) + q(0) = 0 + 0 = 0, so p + q is in H. For any scalar c, (cp)(0) = c * p(0) = c * 0 = 0, so cp is in H. Thus H is a subspace.