# Let Z be a cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eingenvalue lambda.Prove that span(Z) is a T-invariant subspace of V.

Question
Upper Level Math
Let Z be a cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eingenvalue lambda.Prove that span(Z) is a T-invariant subspace of V.

2021-02-16
Here, we are given that Z is the cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eigenvalue $$\displaystyle\lambda$$.
Let T be a linear operator on a vector space , and let be an eigenvalue of . Suppose that
$$\displaystyle{Z}={\left\lbrace{v}_{{1}},{v}_{{2}},\ldots,{v}_{{n}}\right\rbrace}$$
Or if p is the smallest positive integer such that $$\displaystyle{\left({T}-\lambda{I}\right)}^{{p}}{v}={0},{Z}={\left\lbrace{\left({T}-\lambda{I}\right)}^{{{p}-{1}}}{v},{\left({T}-\lambda{I}\right)}^{{{p}-{2}}}{v},\ldots,{\left({T}-\lambda{I}\right)}{v},{v}\right\rbrace}$$,
cycles of generalized eigenvectors of corresponding to .
Step 2
Now, we need to show that span(Z) is a T-invariant subspace of V.
Let us take a vector w from span(Z).
So, we can be written as linear combination of $$\displaystyle{v}_{{1}},{v}_{{2}},\ldots,{v}_{{n}}$$.
i.e., $$\displaystyle{w}=\alpha_{{1}}{v}_{{1}}+α{l}{p}{h}{a}_{{2}}{v}_{{2}}+\ldots+\alpha_{{n}}{v}_{{n}}{w}{h}{e}{r}{e}{a}{t}\le\ast{o}\ne\alpha_{{i}}\ne{0}$$.
Now, it can be observed that if we can show that the vectors are T invariant then w is also T invariant.
$$\displaystyle{T}{\left({v}_{{i}}\right)}={T}{\left({\left({T}-\lambda{I}\right)}^{{k}}{v}\right)}$$, for some k (by the definition)
$$\displaystyle={\left({T}-\lambda{I}+\lambda{I}\right)}{\left({\left({T}-\lambda{I}\right)}^{{k}}{v}\right)}$$
$$\displaystyle={\left({T}-\lambda{I}\right)}{\left({T}-\lambda{I}\right)}^{{k}}{v}+\lambda{I}{\left({T}-\lambda{I}\right)}^{{k}}{v}$$
$$\displaystyle={\left({T}-\lambda{I}\right)}^{{{k}+{1}}}{v}+\lambda{I}{\left({T}-\lambda{I}\right)}^{{k}}{v}$$
$$\displaystyle={\left({T}-\lambda{I}\right)}^{{{k}+{1}}}{v}+{\left({T}-\lambda{I}\right)}^{{k}}{\left(\lambda{v}\right)}$$
And note that $$\displaystyle{\left({T}-\lambda{I}\right)}^{{{k}+{1}}}{v}{\quad\text{and}\quad}{\left({T}-\lambda{I}\right)}^{{k}}{\left(\lambda{v}\right)}$$ are also in span(Z), i.e., T-invariant.
Now, since w was taken randomly, span(Z) is a T-invariant subspace of V.

### Relevant Questions

Find a transformation from u,v space to x,y,z space that takes the triangle $$\displaystyle{U}={\left[\begin{array}{cc} {0}&{0}\\{1}&{0}\\{0}&{1}\end{array}\right]}$$ to the triangle
$$\displaystyle{T}={\left[{\left({1},{0}\right)},-{2}\right)},{\left(-{1},{2},{0}\right)},{\left({1},{1},{2}\right)}{]}$$
A bipolar alkaline water electrolyzer stack module comprises 160 electrolytic cells that have an effective cell area of $$\displaystyle{2}{m}^{{2}}$$. At nominal operation, the current density for a single cell of the electrolyzer stack is 0.40 $$\displaystyle\frac{{A}}{{c}}{m}^{{2}}$$. The nominal operating temperature of the water electrolyzer stack is $$\displaystyle{70}^{\circ}$$ C and pressure 1 bar. The voltage over a single electrolytic cell is 1.96 V at nominal load and 1.78 V at 50% of nominal load. The Faraday efficiency of the water electrolyzer stack is 95% at nominal current density, but at 50% of nominal load, the Faraday efficiency decreases to 80%.
Calculate the nominal stack voltage:
Calculate the nominal stack current:
Calculate the nominal power on the water electrolyzer stack:
The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. Then the volume (in centimeters cubed) of a ball of radius r centimeters is $$\displaystyle\frac{{4}}{{3}}\pi{r}^{{3}}$$. The surface area is $$\displaystyle{4}\pi{r}^{{2}}$$.Set up the differential equation for how r is changing. Then, suppose that at time t = 0 minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time t will the snowball be completely melted.
Consider $$\displaystyle{V}={s}{p}{a}{n}{\left\lbrace{\cos{{\left({x}\right)}}},{\sin{{\left({x}\right)}}}\right\rbrace}$$ a subspace of the vector space of continuous functions and a linear transformation $$\displaystyle{T}:{V}\rightarrow{V}$$ where $$\displaystyle{T}{\left({f}\right)}={f{{\left({0}\right)}}}\times{\cos{{\left({x}\right)}}}−{f{{\left(π{2}\right)}}}\times{\sin{{\left({x}\right)}}}.$$ Find the matrix of T with respect to the basis $$\displaystyle{\left\lbrace{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}},{\cos{{\left({x}\right)}}}−{\sin{{\left({x}\right)}}}\right\rbrace}$$ and determine if T is an isomorphism.
Let S be a subset of an F-vector space V. Show that Span(S) is a subspace of V.
Find a square number such that when twice its root is added to it or subtracted from it, one obtained other square numbers. In other words, solve a problem of the type.
$$\displaystyle{x}^{{2}}+{2}{x}={u}^{{2}}$$
$$\displaystyle{x}^{{2}}-{2}{x}={v}^{{2}}$$
Let $$\displaystyle\le{f}{t}{\left\lbrace{v}_{{{1}}},\ {v}_{{{2}}},\dot{{s}},\ {v}_{{{n}}}{r}{i}{g}{h}{t}\right\rbrace}$$ be a basis for a vector space V. Prove that if a linear transformation $$\displaystyle{T}\ :\ {V}\rightarrow\ {V}$$ satisfies $$\displaystyle{T}{\left({v}_{{{1}}}\right)}={0}\ \text{for}\ {i}={1},\ {2},\dot{{s}},\ {n},$$ then T is the zero transformation.
Getting Started: To prove that T is the zero transformation, you need to show that $$\displaystyle{T}{\left({v}\right)}={0}$$ for every vector v in V.
(i) Let v be an arbitrary vector in V such that $$\displaystyle{v}={c}_{{{1}}}\ {v}_{{{1}}}\ +\ {c}_{{{2}}}\ {v}_{{{2}}}\ +\ \dot{{s}}\ +\ {c}_{{{n}}}\ {v}_{{{n}}}.$$
(ii) Use the definition and properties of linear transformations to rewrite $$\displaystyle{T}\ {\left({v}\right)}$$ as a linear combination of $$\displaystyle{T}\ {\left({v}_{{{1}}}\right)}$$.
(iii) Use the fact that $$\displaystyle{T}\ {\left({v}_{{i}}\right)}={0}$$ to conclude that $$\displaystyle{T}\ {\left({v}\right)}={0}$$, making T the zero tranformation.
Guided Proof Let $${v_{1}, v_{2}, .... V_{n}}$$ be a basis for a vector space V.
Prove that if a linear transformation $$T : V \rightarrow V$$ satisfies
$$T (v_{i}) = 0\ for\ i = 1, 2,..., n,$$ then T is the zero transformation.
To prove that T is the zero transformation, you need to show that $$T(v) = 0$$ for every vector v in V.
(i) Let v be the arbitrary vector in V such that $$v = c_{1} v_{1} + c_{2} v_{2} +\cdots + c_{n} V_{n}$$
(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of $$T(v_{j})$$ .
(iii) Use the fact that $$T (v_{j}) = 0$$
to conclude that $$T (v) = 0,$$ making T the zero transformation.
Using the Mathematical Induction to prove that: $$\displaystyle{3}^{{{2}{n}}}-{1}$$ is divisible by 4, whenever n is a positive integer.