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## Which of the following subsets are subspaces?

As polynomials, they should generally preserve the scalar multiple and addition properties of subspaces. I believe the key is that they need to include the zero vector.

(1) p'(t) = constant

That constant could be zero, and it could be integrated to give another zero constant. It includes the zero vector.

(2) p(-t) = -p(t)

This is the definition of odd functions generally, but a special ‘exception’ exists:

p(t) = 0

p(-t) = 0 = -p(t)

This includes the zero vector.

(3) p(5) = 3

3 can have scalar multiplication and addition applied to it, but it doesn’t include the zero vector.

(4) It is zero.

(5) Again, zero.

(6) The differential equation is zero, but the p function isn’t directly.

If you make p(t) = 0, you get:

p'(t) + 9p(t) + 4 = 0

( 0 ) + 9( 0 ) + 4 = 0

4 = 0

This is not true. The zero vector is not included.

Thus: 1, 2, 4, 5.

Incidentally, the solution of the differential equation should be:

p(t) = c·e^(-9·t) – 4/9

If this were a DE with just exponentials with constants, the constants could both be zero. Then it would include the zero vector. If it were just functions and derivatives like:

4y” + 5y’ + 20y = 0

then it would be a subspace (a linear homogeneous differential equation, incidentally).

Power Set | Subset – The empty set {} is a subset of {a,b,c}. And these are subsets: {a}, {b} and {c}. Think of it as all the different ways we can select the items (the order of the items doesn't matter), including selecting none, or all.Subspaces, subsets, linear independence etc. Advanced Algebra. Mar 22, 2012. Which of the following subsets are subspaces? Differential Geometry. Nov 8, 2011. Help with homework: vector spaces, subspaces, subsets. Advanced Algebra.§3.2 5. Determine whether the following are subspaces of P4. c) The set of all polynomials p(x) in P4 such that p(0) = 0 is a subspace of P4 becuase it satises both conditions of a subspace. To see this rst note that all elements of the set described by (c) can be written in the form p(x) = ax3 + bx2…

Subsets that are subspaces of P2 | Math Help Forum – Why are subspaces important? Well, it turns out a subspace inherits most of the vector space axioms from its parent vector space. Denition: Suppose that V is a vector space. Then U is a subspace of V if is satises the following three properties: S0: U is a non-empty subset of V S1: x + y ∈ U for all x, y…{'transcript': "Okay. The question is, let us be the subspace of pollen. No meals. 32 3/3 degree pollen. So find a set of actors that spans P of X is constant. So I'm going to say that my vector said, contains the element one We know it's not empty because it was one in it.set is a subspace of Pn bc set contains zero vector/closed under addition/closed under scalar mult. 5. Let W be the set of all vectors of the form shown, where a and b represent arbitrary real numbers. Find a set S of vectors that spans W, or give an example or an explanation showing why W is not a vector…

PDF Show that S is a vector space. – Proposition A subset S of a vector space V is a subspace of V if and only if S is nonempty and closed under linear operations, i.e. The span of the set S, denoted Span(S), is the smallest subspace of V that contains S. That is, • Span(S) is a subspace of V ; • for any subspace W ⊂ V one has.This preview shows page 1 – 3 out of 4 pages. 6.(1 point) Which of the following subsets ofP2(R)aresubspaces ofP2(R)?A subspace turns out to be exactly the same thing as a span, except we don't have a particular set of spanning vectors in mind. This change in perspective is quite useful, as it is easy to produce subspaces that are not obviously spans. For example, the solution set of the equation.