Let A be a matrix with independent rows. A. Show that AAT is invertible. B. Show that if b is any vector in Col(A), then the equation Ax = b has a solution in Row(A) given by xR = AT (AAT )−1 b. (The matrix AT (AAT )−1 is called the pseudoinverse of A, and is usually denoted by A+ . It is a right inverse of A, and coincides with the two-sided inverse A−1 if A is square.) c. The mapping of Col(A) to Row(A) given by xR = AT (AAT )−1 b is an isomorphism

Answers

Answer 1

Answer:

Attached below

Step-by-step explanation:

Given that A is a matrix with independent rows

A) Prove [tex]AA^{T}[/tex] is invertible

rank of A = number of rows of Matrix A

This shows that | A | ≠ 0  ( i.e. A has a full rank )

also  [tex]|A^{T} |[/tex] ≠ 0

and [tex]|AA^{T}|[/tex] = | A | [tex]|A^{T} |[/tex]   ≠ 0

Hence we can conclude that [tex]AA^{T}[/tex] is invertible

B) Prove that b is any vector in Col( A )

Attached below is the detailed solution

C) Mapping of Col ( A ) to Row ( A )

b is a non-zero vector hence AXr = 0 ( i.e. there is no solution )

also the kernel of the mapping will be Null

show that mapping preserves the operation

attached below is the detailed solution

Let A Be A Matrix With Independent Rows. A. Show That AAT Is Invertible. B. Show That If B Is Any Vector

Related Questions

The train journey between London King's cross and Durham is 2hours and 50 minutes. If a train leaves London at 08:40 at what time doses it arrive in Durham?

Answers

Answer:

11:30

Step-by-step explanation:

8:40 + 2 hours = 10:40

10:40 + 20 min = 11:00

11:00 + 30 min = 11:30

What is the volume, in cubic m, of a cube with an edge length of 11m?
A rectangular prism has a length of 17m, a height of 18m, and a width of 17m. What is its volume, in cubic m?

Answers

11x11x11=1331 m^3
For the Q
17x18x17=5202 m^3
For the rectangular prism

938.120 in word form please

Answers

Answer:

nine hundred thirty-eight one hundred twenty thousandths

Step-by-step explanation:

Write the equation of the circle with a radius 7 and center (13, 15).


HELP ASAP!!!

Answers

Answer:

Option b is the correct answer

[tex] (x - 13)^2 + (y - 15)^2 = 49 [/tex]

Step-by-step explanation:

[tex] (x - 13)^2 + (y - 15)^2 = 7^2 [/tex]

[tex] (x - 13)^2 + (y - 15)^2 = 49 [/tex]

Martin's bank account balance is negative. Each week that his
account remains negative the bank charges $36.50. Martin did
not make any deposits and now his bank account balance is
-$182.50. How many weeks was his account negative?

Answers

Answer:

5

Step-by-step explanation:

182.50/36.50 per week= 5 weeks

could you help me by chance?

Answers

Answer:

27.5

This is because you plug the numbers into the equation. Hope this helps!

please help like waaa everywhere it comes up that its impossible

Answers

Answer:

Step-by-step explanation:

If we subtract 7 from both sides, we get -2 on the right.  This is not possible, since absolute values are always 0 or greater.

What is the longest line segment that can be drawn in a right rectangular prism that is 12 cm long, 11 cm wide, and 8 cm tall?

Answers

When I put the right answer it says it’s wrong

Answer:

The answer is 18.1

1. The Community Wading Pool is in the shape of a circle. If the radius of the pool is 28 feet, what is the distance around the pool? I need help

Answers

28 feet x 3 == 28373

Answer:

28 feet

Step-by-step explanation:

thank you the same as the guy above me

A child swallowed a substance with an LD50 of 6 mg/kg of body weight. It is crucial that the child's stomach be pumped promptly as the material is highly poisonous.


False

True

Answers

Yes, It is crucial that the child's stomach be pumped promptly as the material is highly poisonous.

How to LD50 represents?

The LD50 is a Lethal dose and 50 represents 50 percent of the test animal's population, that is it will be lethal to half of the tested animals. The amount of a toxic agent (such as a poison, virus, or radiation) that is powered to wipe 50 percent of a population of animals on which it is tested.

Therefore, It is crucial that the child's stomach be pumped promptly as the material is highly poisonous.

Learn more about LD50 here: https://brainly.com/question/10042271

#SPJ2

Please help!! Also please show how to do problem!

Answers

Answer:

118000 thousand cents. Jdbduxb

The manager of a snack bar buys bottled water in packs of 35 and candy bars in packs of 20. Then, she sells the items individually. Yesterday, she bought the same number of water bottles as candy bars to sell at the snack bar. If she bought the lowest number of items possible, how many packs of each item did she buy?

Answers

[tex]\Huge\bf{\underline{\red \dag { Aиѕωєя }}} \red \dag[/tex]

4 Packs of water and 7 Packs of candy bars.

So, we wanna know the smallest number that both 35 and 20 will go into.

Find the Least Common Multiple (LCM), but to find the LCM we need to find the prime factorization of each of the following number :-

[tex] \sf {35 = 5(7)}[/tex]

[tex] \sf \red {20 = 5(4)}[/tex]

[tex] \sf {4 = 2(2)}[/tex]

[tex] \sf \red {20 = 5(2)(2)[/tex]

~Now multiply all the numbers by 5 :-

[tex] \sf \pink{5(7)(2)(2) = 35(2)(2) = 70(2) = 140}[/tex]

This means she needs 140 bottles of water and 140 candybars.

Water is sold in packs of 35, this means that she needs :

[tex] \sf \orange{ \frac{140}{35} = 4\: packs \: of \: water}[/tex]

Candy bars are sold in packs of 20, this means she needs :

[tex] \sf \green{ \frac{140}{20} = 7 \: packs \: of \: candy \: bars}[/tex]

_________________________

Which equation has a constant of proportionality equal to 10? Choose 1 answer: Choose 1 answer:

Answers

it’s B! I hope I get brainlist!! I hope you have a good day bye!






John finished his 5 chores in 45 minutes. If he spent the same amount of time on each
chore, how long, in minutes, did it take him to finish each chore?

Answers

Answer:

9 Minutes

Step-by-step explanation:

5x9=45

Brainliest?

9 minutes 5x9=45…………..

For each sentence, drag a word that will make the statement true.
Words may be used once or more than once.
always returned to choices list.
sometimesalwaysnever
Equilateral triangles are
acute triangles.

Scalene triangles are
acute triangles.

Right triangles are
acute triangles.

Obtuse triangles are
isosceles triangles.

Answers

Answer:

Equilateral triangles are always acute triangles.

Scalene triangles are sometimes acute triangles.

Right triangles are never acute triangles.

Obtuse triangles are sometimes isosceles triangles.

Step-by-step explanation:

Equilateral triangles are always acute because each angle is 60⁰.

Scalene triangles have sides that are different lengths. They can be right, obtuse, or acute.

A right triangle is never acute because it has a 90⁰ angle. Acute means all angles are less than 90⁰.

An obtuse triangle can be either scalene or isosceles. It always has one angle greater than 90⁰ (obtuse).

PLEASE HELP ME ASAP!!! No links please and please give a step by step explanation

Is 547 a term in the sequence 1,8,15,22,29?

Answers

Answer:

no

Step-by-step explanation:

answer: b. f(x) = (x + 2)^3 (x^2 - 7x + 3)^4

step-by-step explanation:

polynomials always have as many roots as their order. this is a 4th order polynomial- hence 4 roots!

complex roots of polynomials always come in pairs. the quadratic equation has the "+/-" in that will alwaysgenerate m+ni amd m-ni, so 7-2i must be a root of this polynomial equation.

the order of the numerator and denominator are both 2. so the horizontal asymptote is y=a2/b2 =7

verticle asymptotes are the forbidden x-values. f(x) becomes undefined at x=-3 because this value will cause a zero denominator.

a graph of r(x) will have vertical asymptotes at ±1 because these values will make the denominator zero. so the domain of the function cannot include ±1.

Urgent Need Help! worth 100 points

Answers

the answer to your question is 4 times 6, 4 times 3 + 4 times 3

have a nice day

Answer:

the answer to this math question is

4 times 6, 4 times 3 + 4 times 3

Step-by-step explanation:

Aleks Verfied

Which shows how to determine the volume of the pyramid?

Answers

Answer:

d would be correct

Step-by-step explanation:

A pyramid is a polyhedron formed by connecting a polygonal base and an a a p e x. The basic formula for pyramid volume is the same as for a cone: volume = (1/3) times  base area times height, where height is the height from the base to the a p e x. That formula is working for any type of base polygon and oblique and right pyramids.

I'll give brainliest​

Answers

Answer:

7 oof

Step-by-step explanation:

What is the distance between the following points?
y
5
3
2
1+
++
-9-8-7 -6 -5 -4 -3 -2
-24
-3
-4+
.

Answers

I don’t know I’m just typing so I can get a Anwser for mine

Answer: c. √65

Step-by-step explanation:

that’s what’s right on khan

Find the area of the circle in terms of Pi. (TT)
10 cm
A

Answers

Answer:

A≈314.16

Step-by-step explanation:

Find the area of the figure.
14 m
5 m
16 m

Answers

Answer:

Step-by-step explanation:

Seperate the shape into two. 16x2=32cm². 12x11=132cm²

So altogether it would = 164cm²

Which angle measure is the measure of a straight angle? 100° 0° 90° 180°

Answers

Answer:

180° degrees is a straight angle

Step-by-step explanation:

Answer:

180 degrees

Step-by-step explanation:

A full circle is 360 degrees

A half circle is a straight angle

a straight angle is 180 degrees

help asap, with good explaination please

Answers

Answer:

60 cubes

Step-by-step explanation:

10 times 6

6 are in each layer but there are 12 going down layer by layer

so 10 times 6 equal 60

the actuall answer is 60

= 60

Simplify -2.3f + 0.8f - 16 - 3

Answers

Answer:

− 1.5 f  −19

that's the correct answer.

Help asap please I really need it!

Answers

Answer:

[tex]volume = \frac{4}{3} \pi r^3\\\\Radius = 2 , volume = \frac{32\pi }{3}cm^3 \\\\Radius = 3, volume = 36\pi cm^3 \\\\Radius = 4, volume = \frac{256\pi }{3} cm^3\\\\Radius= 6, volume = 288\pi cm^3[/tex]

what is the 12th term of the sequence: 48,44,40,36...

Answers

Answer:

12

Step-by-step explanation:

The sequence starts at 48, and goes down 4 with each term. That means that we just need to subtract 4 from 48 12 times.

Hope that this helps!

Please help me with this math problem

Answers

Answer:

80.6 m

Step-by-step explanation:

Coordinates of the points P and Q are (0, 0) and (40, 70) respectively.

Length of the shortcut can be found by using distance formula between two points P and Q.

Length of the shortcut

[tex] = \sqrt{ {(40 - 0)}^{2} + {(70 - 0)}^{2} } \\ \\ = \sqrt{ {40}^{2} + {70}^{2} } \\ \\ = \sqrt{1600 + 4900} \\ \\ = \sqrt{6500} \\ \\ = 80.6225775 \\ \\ \approx \: 80.6 \: m[/tex]

45 yd
24 yd
What is the length of the hypotenuse?

Answers

Answer= 51
I plugged it into a calculator good luck

Answer:

51

Step-by-step explanation:

A^2+b^2=c^2

That's the formula

Complete the equation so it will have no solution. 5x - 8 = _ +2

Answers

Answer:

5x

Step-by-step explanation:

If the blank were to be occupied with 5x there would be no solutions

Observe...

5x - 8 = 5x + 2

* subtract 5x from each side *

5x - 5x cancels out

5x - 5x cancels out

we're left with -8 = 2

which is not true

Hence if the blank = 5x the equation will have no solution

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