# 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.

Question
Transformation properties
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.

2021-01-18
a)Given:
The linear transformation $$T : V \rightarrow V$$
represented as $$T (v_{i}) = 0\ for\ i = 1, 2, ..., n.$$
Approach:
Consider an arbitrary $$v = {v_{1}, v_{2},..., v_{n}}$$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$$T(v_{i}) = 0, i = 1, 2, \cdots,n...(1)$$
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
$$v = c_{1} v_{1} + c_{2} v_{2} + \cdots + c_{n} v_{n} \cdots, (2)$$
Here, $$c_{1}, c_{2}, \cdots c_{n}$$ are arbitrary scalars.
Conclusion:
Hence, it is proved above that the set $$(v_{1}, v_{2}, \cdots v_{n}}$$ is represented as
$$v = c_{1} n_{1} + c_{2} v_{2}, + \cdots + c_{n} v_{n}.$$
b)Given:
The linear transformation $$T : V \rightarrow V$$
represented as $$T (v_{i}) = 0\ for\ i = 1, 2, \cdots, n.$$
Approach:
Consider an arbitrary $$v = {v_{1}, v_{2},\cdots, v_{n}}$$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$$T(v_{i}) = 0, i = 1, 2, \cdots,n \cdots(1)$$
The vector additinon is given by,
$$T (u + v) = T (u) + T (v)$$
The scalar multiplication is given by,
$$T (cu) = cT (u)$$
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
$$T(v) = (c_{1} v_{1} + c_{2} v_{2} + \cdots + c_{n} v_{n})$$
$$= T(c_{1} v_{1}) + T(c_{2} v_{2}) + \cdots + T (c_{n} v_{n})$$
$$= c_{1} T(v_{1}) + c_{2} T( v_{2}) + \cdots + c_{n} T(v_{n})....(3)$$
Conclusion:
The transformation form of linear combination $$v = c_{1} v_{1} + c_{2} v_{2} + \cdots + c_{n} v_{n}$$ is
$$T(v) = c_{1} T(v_{1}) + c_{2} T( v_{2}) + \cdots + c_{n} T(v_{n})$$
c) Given:
The linear transformation $$T : V \rightarrow V$$
represented as $$T (v_{i}) = 0\ for\ i = 1, 2, \cdots, n.$$
Approach:
Consider an arbitrary $$v = {v_{1}, v_{2},\cdots, v_{n}}$$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$$T(v_{i}) = 0, i = 1, 2, \cdots,n \cdots(1)$$
The vector additinon is given by,
$$T(u + v) = T(u) + T(v)$$
The scalar multiplication is given by,
$$T(cu) = cT(u)$$
Calculation:
Solve formula (3) with use of formula(1)
$$T(v) = c_{1} T(v_{1}) + c_{2} T( v_{2}) + \cdots + c_{n} T(v_{n})$$
$$= c_{1} (0) + c_{2} (0) + \cdots + c_{n} (0)$$
= 0
From above calculation is is clear linear transformation $$T : V \rightarrow V$$
satisfies $$T (v_{i}) = 0\ for\ i = 1, 2, \cdots, n,$$ than T is the zero transformation.
Conclusion:
Hence, ithe solution of $$T(v) = c_{1} T(v_{1}) + c_{2} T( v_{2}) + \cdots + c_{n} T(v_{n})$$ is 0 which shows that T is zero transformation.

### Relevant Questions

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.

Let $$\left\{v_{1},\ v_{2}, \dots,\ v_{n}\right\}$$ 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.

Let $$T : U \rightarrow U$$ be a linear transformation and let beta be a basis of U Define the determinant det(T) of T as
det$$(T) = det([T]_{\beta}).$$
Show ta det (T) is well-defined, i. e. that it does not depend on the choice of the basis beta
Prove that T is invertible if and only if det $$(T) \neq 0.$$ If T is invertible, show that det $$(T^{-1}) = \frac{1}{det(T)}$$
Let $$X_{1}....,X_{n} and Y_{1},...,Y_{m}$$ be two sets of random variables. Let $$a_{i}, b_{j}$$ be arbitrary constant.
Show that
$$Cov(\sum_{i=1}^{n}a_{i}X_{i},\sum_{j=1}^{m}b_{j}Y_{j})=\sum_{i=1}^{n}\sum_{j=1}^{m}a_{i}b_{j}Cov(X_{i}, Y_{j})$$
To find:
The linear transformation $$\displaystyle{\left({T}_{{2}}{T}_{{1}}\right)}{\left({v}\right)}$$ for an arbitrary vector v in V.
The vectors $$\displaystyle{\left\lbrace{v}_{{1}},{v}_{{2}}\right\rbrace}$$ is vasis for the vector space V.
Given:
The linear transformation with satisfying equations $$\displaystyle{T}_{{{1}}}{\left({v}_{{1}}\right)}={3}{v}_{{{1}}}+{v}_{{{2}}},$$
$$\displaystyle{T}_{{{1}}}{\left({v}_{{1}}\right)}=-{3}{v}_{{{1}}}+{v}_{{{2}}},\ {T}_{{{2}}}{\left({v}_{{{1}}}\right)}=-{5}{v}_{{{2}}},$$ and $$\displaystyle{T}_{{{2}}}{\left({v}_{{2}}\right)}=-{v}_{{{1}}}+{6}{v}_{{{2}}}$$ are given as
$$\displaystyle{T}_{{{1}}}\ :\ {V}\ \rightarrow{V}$$ and $$\displaystyle{T}_{{{2}}}\ :\ {V}\rightarrow{V}.$$
Let $$T\ :\ U\ \rightarrow\ U$$ be a linear transformatiom and let
$$\mathscr{B}$$ be a basis of U. Define the determanant det (T) of T as
$$det(T) = det([T]\ \mathscr{B}).$$
Show that det (T) is well=defined, i.e. that it does not depend on the choice of the basis $$\mathscr{B}.$$
Prove that T is invertible if and only if det $$(T) \neq\ 0.$$ If T is invertible, show that
$$det(T^{-1}) =\ \frac{1}{det(T)}.$$
Assum T: R^m to R^n is a matrix transformation with matrix A. Prove that if the columns of A are linearly independent, then T is one to one (i.e injective). (Hint: Remember that matrix transformations satisfy the linearity properties.
Linearity Properties:
If A is a matrix, v and w are vectors and c is a scalar then
$$A 0 = 0$$
$$A(cv) = cAv$$
$$A(v\ +\ w) = Av\ +\ Aw$$
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
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.
The set $$\displaystyle{\left\lbrace{T}{\left({x}_{{1}}\right)},\ \ldots\ ,{T}{\left({x}_{{k}}\right)}\right\rbrace}$$ is a linearly independent subset of $$\displaystyle{R}^{{{m}}}$$
Let $$\displaystyle{T}\ :\ {T}\ :\ {R}^{{{n}}}\rightarrow{R}^{{{m}}}$$ be a linear transformation with nulity zero. If $$\displaystyle{S}={\left\lbrace{x}_{{{1}}},\ \cdots\ \ ,{x}_{{{k}}}\right\rbrace}$$ is a linearly independent subset of $$\displaystyle{R}^{{{n}}}.$$