linear algebra proof question please show clear thanks Problem 3: Let T: V → V be a linear map, let v € V, and let W be the T-cyclicsubspace of V generated by v.a.) For any u € V, show that u € W if and only if there exists a polynomial f(t)such that u = f(T)v.b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k.

Given that T:V→V be a linear map and v∈V. Given that W be the T-cyclic subspace generated by v. We…

Answered: Problem 3: Let T: V → V be a linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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linear algebra proof question

please show clear thanks

Problem 3: Let T: V → V be a linear map, let v € V, and let W be the T-cyclic
subspace of V generated by v.
a.) For any u € V, show that u € W if and only if there exists a polynomial f(t)
such that u = f(T)v.
b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

Expert Solution

Step 1

Given that $T : V \to V$ be a linear map and $v \in V$ .

Given that $W$ be the $T -$ cyclic subspace generated by $v$ .

We know that a subspace $W$ is a $T -$ cyclic subspace generated by $v$ if:

$W = span \{v, T (v), T^{2} (v), \dots\}$ .

(a)

Let $u \in V$ be an arbitrary element.

Forward proof:

Suppose that $u \in W$ .

Since $W$ be the $T -$ cyclic subspace generated by $v$ and hence $W = span \{v, T (v), T^{2} (v), \dots\}$ , therefore $u \in span \{v, T (v), T^{2} (v), \dots\}$ .

Therefore there exists a largest natural number $n$ and $c_{0}, c_{1}, \dots, c_{n} \in F$ such that:

$\begin{array}{rcl} u & = & c_{0} v + c_{1} T (v) + c_{2} T^{2} (v) + \dots + c_{n} T^{n} (v) \\ = & (c_{0} I + c_{1} T + c_{2} T^{2} + \dots + c_{n} T^{n}) (v) \end{array}$ .