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
dimulka [17.4K]
2 years ago
13

Use mathematical induction to prove that if L is a linear transformation from V to W, then L (α1v1 + α2v2 +· · ·+αnvn)= α1L (v1)

+ α2L (v2)+· · ·+αnL (vn) g
Mathematics
2 answers:
e-lub [12.9K]2 years ago
6 0

Answer:

The proof is shown in the explanation below.

Step-by-step explanation:

Analysis:

The proof by induction focuses on n. In this case, let n = 1, and L^{1} will be a linear operator since L^{1} = L

The exercise will show that L^{n} is a linear operator on V and that L^{n+1} is also a linear operator on V. This, follows that:

L^{n+1} (av) = L(L^{m}(v_{1}+v_{2})\\                   = L(L^{m} (v_{1} + L^{m}v_{2})\\                   = L(L^{m}v_{1} + L(L^{m}v_{2})\\                   = L^{m+1}(v_{1}) + L^{m+1}(v_{2})

Kay [80]2 years ago
4 0

Answer/Step-by-step explanation:

For the mathematical induction,

We show that the equation

L (α1v1 + α2v2 +· · ·+αnvn)= α1L (v1) + α2L (v2)+· · ·+αnL (vn) is true for

L = 1,

Assume it is true for L = n and show that it is true for L = n + 1.

If L = 1, the equation become

(α1v1 + α2v2 +· · ·+αnvn)= α1(v1) + α2 (v2)+· · ·+αn (vn). Therefore, the Right Hand side(RHS) = Left Hand side(LHS)

When L = n, we assume the following is true

(α1nv1 + α2nv2 +· · ·+αnvn)= α1n(v1) + α2n (v2)+· · ·+αn (vn)

Then, when L = n + 1,

n +1 (α1v1 + α2v2 +· · ·+αvn)= α1(n +1) (v1) + α2(n +1) (v2)+· · ·+αn(n + 1)(vn).

Open the bracket,

n(α1v1 + α2v2 +· · ·+αvn) + α1v1 + α2v2 +· · ·+αnvn = α1n (v1) + α2v2 +· · ·+αvn ) + α1(v1) + α2v2+· · ·+αn(vn)

Since we assume the the equation is true for L = n, and eliminating some terms, then

L (α1v1 + α2v2 +· · ·+αnvn)= α1L (v1) + α2L (v2)+· · ·+αnL (vn)

You might be interested in
32 3/4 take away 12 1/2​
Otrada [13]

Answer:

The answer is 20 and 1/4

7 0
3 years ago
Read 2 more answers
Eight more than the product of some number cubed and two. What does that look like written out?
eduard
The Answer is 8+(x^3)*2
3 0
3 years ago
A three-dimensional model of lucia’s apartment is shown. to cool the whole apartment, an air conditioner must provide 2.5 btus p
SashulF [63]

The number of btus which are needed to cool Lucia’s apartment shown in the three-dimensional model are 20,400 btus.

<h3>What is of volume of rectangular prism?</h3>

Volume of rectangular prism is the amount of quantity, which is obtained in it in the 3 dimensional space.

Volume of a rectangular prism is the product of its width, height and length.

V=w\times h\times l

Here, (w) is the width of the base (l) is the length and (h) is the height of the rectangular prism.

A three-dimensional model of Lucia’s apartment is shown. The image is attached below for the apartment. The three section are there in the apartment. The volume of this apartment is,

V=[(20\times12)+(20-9)\times16+(20\times20)]\times10\\V=8160\rm\; ft^3

To cool the whole apartment, an air conditioner must provide 2.5 btus per cubic foot. To cool, the 8160 cubic foot apartment the btus needed are,

n=8160\times2.5\\n=20400\rm\; btus

Hence, the number of btus which are needed to cool Lucia’s apartment shown in the three-dimensional model are 20,400 btus.



Learn more about the volume of a rectangular prism here;

brainly.com/question/4016105

5 0
2 years ago
A number increased by 24 is 112. what is the number
dexar [7]
I believe the correct number is 88.
8 0
2 years ago
Write an equation in point slope form of the he that passes through the point (2.1) and has a slope of 2
Wewaii [24]

Answer:

y=2x-3

Step-by-step explanation:

show work

1=(2*2)+b

1=4+b

1-4=-3

-3=b

check work

y=2x-3

y=(2*2)-3

y=4-3

y=1

5 0
3 years ago
Read 2 more answers
Other questions:
  • Alisa hopes to play beach volleyball in the Olympics someday. She has convinced her parents to allow her to set up a beach volle
    8·2 answers
  • I need help please and thank you
    13·1 answer
  • To solve the system of equations below, Maria isolated the variable y in the first equation and then substituted it into the sec
    13·2 answers
  • Suppose that a coin has been altered to come up heads 60% of the time. If many samples of 60 coin flips are taken, which of the
    7·1 answer
  • A heron can travel an average of 400 km in 10h. What is the rate of change of distance?
    15·2 answers
  • Domain and Range NAME: MR. Q State the domain and range for each graph and then tell if the graph is a function (write yes or no
    6·1 answer
  • A restaurant bill is 56.39.Estimate a 15% tip
    6·1 answer
  • Random khan question and I haven't done this in a year, I'm struggling.
    11·1 answer
  • Dan lives in Pittsburgh and takes a plane to Miami. When Dan left the house it was -10°. When the plane landed in Miami, Dan rea
    12·1 answer
  • What percent of 93 is 22?
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!