- #1
- 257
- 17
- Homework Statement
- 1. Let ##F## (##\mathbb{R}## or ##\mathbb{C}##) be a field, let ##T## be a nonempty set, and let ##V## be the set of all functions ##x: T \rightarrow F##. ##~V = \mathcal{F} (T, F)## is a vector space of linear functions. In the case that ##T = \{1, 2, ..., n\}## for some positive integer ##n##, do you see any similarity between ##V## and ##F^n##?
2. Let ##F## be a field and let ##V## be the set of all 'infinite sequences' ##x = (a_1, a_2, a_3, ...)## of elements of ##F##. It is clear that ##V## is a vector space over ##F##. Do you see any similarity between ##V## and the space ##\mathcal{F} (\mathbb{Z}^+, F)## obtained by setting ##T = \mathbb{Z}^+## as in problem 1?
3. Let ##F## be a field. If ##p## is a polynomial with coefficients in ##F##, regard ##p## as a function on ##F## such that,
$$p(t) = a_0 + a_1 t + a_2 t^2 + . . . + a_n t^n$$
where ##a_i##'s and ##t## are in ##F##. So we can define ##p: F \rightarrow F## where ##p \in \mathcal{P}##. It is clear that ##\mathcal{P}## is a vector space.
A minor variation of problem 2: consider sequences indexed by the set ##N = \{0, 1, 2, 3, ...\}## of nonnegative integers, that is, sequences
##x =(a_0, a _1, a_2, ...)## with ##a_i \in F## for all ##i \in N##. This notation is better adapted for the space of polynomials ##\mathcal{P}##, explain.
4. Let ##V## be the space of infinite sequences ##x =(a_0, a _1, a_2, ...)## indexed by ##N = \{0, 1, 2, 3, ...\}##. Consider the set ##W## of all ##x \in V## such that, from some index onward, the ##a_i## are all ##0##, i.e. ##x = (0, -3, 1, 5, 0, 0, 0, ...)##. It is easy to see that ##W \subset V##. Do you see any similarity between ##W## and the space ##\mathcal{P}## of polynomial functions?
- Relevant Equations
- Concepts about functions
Vector space axioms
Solution
1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##.
From this, we can see that we can form a correspondence between ##x(1)## and the first element of ##(x_1, x_2, ..., x_n)## which is ##x_1##. So we can say that ##x(1) = x_1##. As a result ##x: T \rightarrow F## is really a map that gives the values of the components of the elements of ##F^n##. I'm not sure if this is correct.
A natural question that arose in my mind, what is the difference between a vector space ##V## of vector valued functions of a real variable ##\vec{v}: \mathbb{R} \rightarrow F^n## and the vector space ##F^n## with element ##\vec{v} = (v_1, v_2, ..., v_n)##? The first one is a vector space of linear maps ##\vec{v}##. The second one is just a vector space with elements ##\vec{v}##.
2. I think this is exactly the same as problem 1, where here ##x(1) = a_1##, ##x(2) = a_2##, and so on. So that here, ##x: \mathbb{Z}^+ \rightarrow F## is the map that gives the values of each component of the element of ##V##.
3. I think we can define a series ##p## as a map ##p: V \times V \rightarrow F## where ##x,y \in V##, ##x =(a _1, a_2, ...)##, and ##y =(t, t^2, ...)## for some ##t \in F## such that,
$$p = x \cdot y = a_1 t + a_2 t^2 + . . . $$
which is an infinite series and can be modified (see problem 4) in order to construct a polynomial. I'm not sure about this.
4. Well, same as in problem 3, although modify the quantities such that ##x =(a_0, a _1, a_2, ..., a^n)##, and ##y =(0, t, t^2, ..., t^n)## and we have
$$p = x \cdot y = a_0 + a_1 t + a_2 t^2 + . . . + a_n t^n$$
but I cannot see the similarity between ##W## and ##\mathcal{P}##, the only thing similar is their result, which is a polynomial.
1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##.
From this, we can see that we can form a correspondence between ##x(1)## and the first element of ##(x_1, x_2, ..., x_n)## which is ##x_1##. So we can say that ##x(1) = x_1##. As a result ##x: T \rightarrow F## is really a map that gives the values of the components of the elements of ##F^n##. I'm not sure if this is correct.
A natural question that arose in my mind, what is the difference between a vector space ##V## of vector valued functions of a real variable ##\vec{v}: \mathbb{R} \rightarrow F^n## and the vector space ##F^n## with element ##\vec{v} = (v_1, v_2, ..., v_n)##? The first one is a vector space of linear maps ##\vec{v}##. The second one is just a vector space with elements ##\vec{v}##.
2. I think this is exactly the same as problem 1, where here ##x(1) = a_1##, ##x(2) = a_2##, and so on. So that here, ##x: \mathbb{Z}^+ \rightarrow F## is the map that gives the values of each component of the element of ##V##.
3. I think we can define a series ##p## as a map ##p: V \times V \rightarrow F## where ##x,y \in V##, ##x =(a _1, a_2, ...)##, and ##y =(t, t^2, ...)## for some ##t \in F## such that,
$$p = x \cdot y = a_1 t + a_2 t^2 + . . . $$
which is an infinite series and can be modified (see problem 4) in order to construct a polynomial. I'm not sure about this.
4. Well, same as in problem 3, although modify the quantities such that ##x =(a_0, a _1, a_2, ..., a^n)##, and ##y =(0, t, t^2, ..., t^n)## and we have
$$p = x \cdot y = a_0 + a_1 t + a_2 t^2 + . . . + a_n t^n$$
but I cannot see the similarity between ##W## and ##\mathcal{P}##, the only thing similar is their result, which is a polynomial.