1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ahrayia [7]
3 years ago
13

Assume {v1, . . . , vn} is a basis of a vector space V , and T : V ------> W is an isomorphism where W is another vector spac

e. Show that the vectors w1 = T(v1), . . . ,wn = T(vn) are a basis of W.
This statement becomes false if we drop the assumption that T is invertible. Demonstrate this by finding a counterexample.
Mathematics
1 answer:
Degger [83]3 years ago
8 0

Answer:

Step-by-step explanation:

To prove that w_1,\dots w_n form a basis for W, we must check that this set is a set of linearly independent vector and it generates the whole space W. We are given that T is an isomorphism. That is, T is injective and surjective. A linear transformation is injective if and only if it maps the zero of the domain vector space to the codomain's zero and that is the only vector that is mapped to 0. Also, a linear transformation is surjective if for every vector w in W there exists v in V such that T(v) =w

Recall that the set w_1,\dots w_n is linearly independent if and only if  the equation

\lambda_1w_1+\dots \lambda_n w_n=0 implies that

\lambda_1 = \cdots = \lambda_n.

Recall that w_i = T(v_i) for i=1,...,n. Consider T^{-1} to be the inverse transformation of T. Consider the equation

\lambda_1w_1+\dots \lambda_n w_n=0

If we apply T^{-1} to this equation, then, we get

T^{-1}(\lambda_1w_1+\dots \lambda_n w_n) =T^{-1}(0) = 0

Since T is linear, its inverse is also linear, hence

T^{-1}(\lambda_1w_1+\dots \lambda_n w_n) = \lambda_1T^{-1}(w_1)+\dots +  \lambda_nT^{-1}(w_n)=0

which is equivalent to the equation

\lambda_1v_1+\dots +  \lambda_nv_n =0

Since v_1,\dots,v_n are linearly independt, this implies that \lambda_1=\dots \lambda_n =0, so the set \{w_1, \dots, w_n\} is linearly independent.

Now, we will prove that this set generates W. To do so, let w be a vector in W. We must prove that there exist a_1, \dots a_n such that

w = a_1w_1+\dots+a_nw_n

Since T is surjective, there exists a vector v in V such that T(v) = w. Since v_1,\dots, v_n is a basis of v, there exist a_1,\dots a_n, such that

a_1v_1+\dots a_nv_n=v

Then, applying T on both sides, we have that

T(a_1v_1+\dots a_nv_n)=a_1T(v_1)+\dots a_n T(v_n) = a_1w_1+\dots a_n w_n= T(v) =w

which proves that w_1,\dots w_n generate the whole space W. Hence, the set \{w_1, \dots, w_n\} is a basis of W.

Consider the linear transformation T:\mathbb{R}^2\to \mathbb{R}^2, given by T(x,y) = T(x,0). This transformations fails to be injective, since T(1,2) = T(1,3) = (1,0). Consider the base of \mathbb{R}^2 given by (1,0), (0,1). We have that T(1,0) = (1,0), T(0,1) = (0,0). This set is not linearly independent, and hence cannot be a base of \mathbb{R}^2

You might be interested in
Pls help whoever help me with all of these will get a brainlist
goldfiish [28.3K]
These are the answers that i got
5 0
3 years ago
Read 2 more answers
A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the
nydimaria [60]

Answer:

Step-by-step explanation:

As you can see, this is a quadratic equation and if you were to graph this you would see that it is a parabola that opens down because of the negative leading coefficient (-34). The maximum profit would be at the vertex, of the graph. Therefore, we need to find the value of "y" when "x" is the line of symmetry. We find this by  x=-b/(2a) where a = -34 and b = 1542:

x=-1542/(2)(-34)

x=-152/-68

x=152/68  

We put this value of "x" into the original formula to find out the value of "y":

y=-34x^2+1542x-10037

y=(-34)(1542/68)^2+1542(1542/68)-10037

y= -17483.5588235 + 34967.1176471 -10037

y=7446.5588236

Therefore, the maximum profit would be $7446.56.

Hope this helps!

PLS MARK BRAINLIEST THIS TOOK A LOT OF TIME

7 0
3 years ago
This problem uses the teengamb data set in the faraway package. Fit a model with gamble as the response and the other variables
hichkok12 [17]

Answer:

A. 95% confidence interval of gamble amount is (18.78277, 37.70227)

B. The 95% confidence interval of gamble amount is (42.23237, 100.3835)

C. 95% confidence interval of sqrt(gamble) is (3.180676, 4.918371)

D. The predicted bet value for a woman with status = 20, income = 1, verbal = 10, which shows a negative result and does not fit with the data, so it is inferred that model (c) does not fit with this information

Step-by-step explanation:

to)

We will see a code with which it can be predicted that an average man with income and verbal score maintains an appropriate 95% CI.

attach (teengamb)

model = lm (bet ~ sex + status + income + verbal)

newdata = data.frame (sex = 0, state = mean (state), income = mean (income), verbal = mean (verbal))

predict (model, new data, interval = "predict")

lwr upr setting

28.24252 -18.51536 75.00039

we can deduce that an average man, with income and verbal score can play 28.24252 times

using the following formula you can obtain the confidence interval for the bet amount of 95%

predict (model, new data, range = "confidence")

lwr upr setting

28.24252 18.78277 37.70227

as a result, the confidence interval of 95% of the bet amount is (18.78277, 37.70227)

b)

Run the following command to predict a man with maximum values ​​for status, income, and verbal score.

newdata1 = data.frame (sex = 0, state = max (state), income = max (income), verbal = max (verbal))

predict (model, new data1, interval = "confidence")

lwr upr setting

71.30794 42.23237 100.3835

we can deduce that a man with the maximum state, income and verbal punctuation is going to bet 71.30794

The 95% confidence interval of the bet amount is (42.23237, 100.3835)

it is observed that the confidence interval is wider for a man in maximum state than for an average man, it is an expected data because the bet value will be higher than the person with maximum state that the average what you carried s that simultaneously The, the standard error and the width of the confidence interval is wider for maximum data values.

(C)

Run the following code for the new model and predict the answer.

model1 = lm (sqrt (bet) ~ sex + status + income + verbal)

we replace:

predict (model1, new data, range = "confidence")

lwr upr setting

4,049523 3,180676 4.918371

The predicted sqrt (bet) is 4.049523. which is equal to the bet amount is 16.39864.

The 95% confidence interval of sqrt (wager) is (3.180676, 4.918371)

(d)

We will see the code to predict women with status = 20, income = 1, verbal = 10.

newdata2 = data.frame (sex = 1, state = 20, income = 1, verbal = 10)

predict (model1, new data2, interval = "confidence")

lwr upr setting

-2.08648 -4.445937 0.272978

The predicted bet value for a woman with status = 20, income = 1, verbal = 10, which shows a negative result and does not fit with the data, so it is inferred that model (c) does not fit with this information

4 0
3 years ago
HELP I need help.<br> The winner will have 3035 points but for now, 100 points.
OLga [1]

Answer:

$6960

Step-by-step explanation:

p is not the principle. It is the principal.

I = prt

p = $6000

r = 8% = 0.08

t = 2

I = ($6000)(0.08)(2)

I = $960

Total amount after 2 years = principal + interest

Total amount = $6000 + $960

Total amount = $6960

8 0
2 years ago
Read 2 more answers
Easy points :) <br> 2*2*2+2-1
kipiarov [429]

Answer:

9

Step-by-step explanation:

2*2=4*2=8+2=10-1=9

7 0
3 years ago
Other questions:
  • Help<br><br> Best answer I will mark as brainliest
    8·2 answers
  • What is greather 1 mi 2,000 yd
    12·1 answer
  • Planes X and Y are perpendicular. Points A, E, F, and G are points only in plane X. Points R and S are points in both planes X a
    9·2 answers
  • polk elementary school was putting on its annual fall carnival. the school was selling 10 tickets for $1.00 or 2 tickets for 25c
    13·1 answer
  • What is the opposite value of 4?
    12·2 answers
  • The values in the table represent a linear function. what is the common difference of the associated arithmetic sequence?
    12·2 answers
  • Help!!!!!!!<br> I will make you brainliest
    6·1 answer
  • What is the area of this circle?
    7·1 answer
  • Find the value of the expression
    12·1 answer
  • I'm desperate, this is my last assignment to do for now​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!