Yasmin
2021-01-05
Answered

Let S be a subset of an F-vector space V. Show that Span(S) is a subspace of V.

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smallq9

Answered 2021-01-06
Author has **106** answers

Theorem. For any set of vectors

Proof. Let

Then there exist

and

Note that

Thus span (S) is closed under addition. M1)

This hosws that span (S) is closed under scalar multiplication. Hence , span (S) is a subspace of V.

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Given basis $B:{b}_{1}={\left(1\text{}2\right)}^{T}$ and $b}_{2}={\left(2\text{}1\right)}^{T$ and $A:{a}_{1}={\left(1\text{}2\right)}^{T},\text{}{a}_{2}={\left(2\text{}7\right)}^{T}$ . Find the transformation matrix T from the basis B into basis A?

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So, I can model growth and decay if I start with assuming that the growth rate is constant:

$\frac{{p}^{\prime}(t)}{p(t)}=\alpha $

and then I have

${p}^{\prime}(t)-\alpha p(t)=0$

A general linear differential equation, however, would have the form

${p}^{\prime}(t)-g(t)p(t)=h(t)$

So the growth rate is g(t). What is h(t)?

And what type of thing is this form of equation used to model?

$\frac{{p}^{\prime}(t)}{p(t)}=\alpha $

and then I have

${p}^{\prime}(t)-\alpha p(t)=0$

A general linear differential equation, however, would have the form

${p}^{\prime}(t)-g(t)p(t)=h(t)$

So the growth rate is g(t). What is h(t)?

And what type of thing is this form of equation used to model?