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
Feliz [49]
3 years ago
9

Please determine whether the set S = x^2 + 3x + 1, 2x^2 + x - 1, 4.c is a basis for P2. Please explain and show all work. It is

fine to use technology.
Mathematics
1 answer:
ohaa [14]3 years ago
5 0

The vectors in S form a basis of P_2 if they are mutually linearly independent and span P_2.

To check for independence, we can compute the Wronskian determinant:

\begin{vmatrix}x^2+3x+1&2x^2+x-1&4\\2x+3&4x+1&0\\2&4&0\end{vmatrix}=4\begin{vmatrix}2x+3&4x+1\\2&4\end{vmatrix}=40\neq0

The determinant is non-zero, so the vectors are indeed independent.

To check if they span P_2, you need to show that any vector in P_2 can be expressed as a linear combination of the vectors in S. We can write an arbitrary vector in P_2 as

p=ax^2+bx+c

Then we need to show that there is always some choice of scalars k_1,k_2,k_3 such that

k_1(x^2+3x+1)+k_2(2x^2+x-1)+k_34=p

This is equivalent to solving

(k_1+2k_2)x^2+(3k_1+k_2)x+(k_1-k_2+4k_3)=ax^2+bx+c

or the system (in matrix form)

\begin{bmatrix}1&1&0\\3&1&0\\1&-1&4\end{bmatrix}\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}a\\b\\c\end{bmatrix}

This has a solution if the coefficient matrix on the left is invertible. It is, because

\begin{vmatrix}1&1&0\\3&1&0\\1&-1&4\end{vmatrix}=4\begin{vmatrix}1&2\\3&1\end{vmatrix}=-20\neq0

(that is, the coefficient matrix is not singular, so an inverse exists)

Compute the inverse any way you like; you should get

\begin{bmatrix}1&1&0\\3&1&0\\1&-1&4\end{bmatrix}^{-1}=-\dfrac1{20}\begin{bmatrix}4&-8&0\\-12&4&0\\-4&3&-5\end{bmatrix}

Then

\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}1&1&0\\3&1&0\\1&-1&4\end{bmatrix}^{-1}\begin{bmatrix}a\\b\\c\end{bmatrix}

\implies k_1=\dfrac{2b-a}5,k_2=\dfrac{3a-b}5,k_3=\dfrac{4a-3b+5c}{20}

A solution exists for any choice of a,b,c, so the vectors in S indeed span P_2.

The vectors in S are independent and span P_2, so S forms a basis of P_2.

You might be interested in
using compound interest, how long does it take to double a 1000 dollar investment that pays 6.5% annual interest, compounded mon
KonstantinChe [14]

Answer:

11 years approx

Step-by-step explanation:

Given data

P=$1000

A=2000

R=6.5%

T= ?

Calculate time, solve for t

t = ln(A/P) / r

substitute

t=ln(2000/1000)/0.065

t=ln(2)/0.065

t=0.693/0.065

t=10.66

Hence the time is 11 years approx

3 0
3 years ago
Consider this expression.
USPshnik [31]

Answer:

-16

Step-by-step explanation:

m^2 + n^2

-5^2 + 3^2 = -16

5 0
1 year ago
What is the largest x where f(x) is discontinuous<br> show work
natulia [17]

Answer:

x = 3

Step-by-step explanation:

reverse foil the denominator and find the largest solution.

x-3(x+1)

3 is the largest solution

6 0
2 years ago
If f(x) = 0.8(2 – x), what is the value of f(–2)?
Sergio039 [100]
With functions, you take the number that is in the ( ), so we have f(-2), and take the fomula f(x) = 0.8(2 – x) and everywhere you see an 'x' replace it with the -2 f(–2)=0.8(2 – (-2)) 

We will have to work with the expression of 0.8(2-(-2) When you want to evaluate these types of expressions, you want to use the Order of Operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. First we have to evaluate the parentheses. What is 2-(-2)
 its 4.

ok now you got 0.8 (4)

 Now we have the remains of 0.8(4). If a number is within the parenthesis, then it means that we have to multiply the number inside with the number that is outside. What is 0.8*4 its 3.2 soooo your answer is going to be

3.2
5 0
3 years ago
<img src="https://tex.z-dn.net/?f=%285y%20%2B%209%29%286y%20-%201%29" id="TexFormula1" title="(5y + 9)(6y - 1)" alt="(5y + 9)(6y
mestny [16]
Hey there :)

( 5y + 9 )( 6y - 1 )

We need to use FOIL to expand, that is
First Terms
Outer Terms
Inner Terms
Last Terms

     First          Outer         Inner      Last
( 5y )( 6y ) + ( 5y )( - 1 ) + 9 ( 6y ) + 9 ( - 1 )
    30y²     -        5y      +    54y   -       9

Combine, if any, the like-terms
30y² + 49y - 9
7 0
3 years ago
Read 2 more answers
Other questions:
  • Suppose customer arrivals at a post office are modeled by a Poisson process N with intensity λ &gt; 0. Let T1 be the time of the
    12·1 answer
  • Find the explicit formula for the geometric sequence cn given below. Note that c1=6.
    13·1 answer
  • Write 3.088 as a mixed number and improper fraction
    6·1 answer
  • 40+5-9x+4<br> I need help with this it needs to be simplified
    7·1 answer
  • Three consecutive integers add up to more than 33. What is the smallest set of integers they could be?
    11·1 answer
  • David earns $7.75 per hour working at a local fast food restaurant. Last week David worked 19.2 hours. How much money did David
    11·1 answer
  • Which shows one way the equation can be represented in words? StartFraction 5 over 8 EndFraction z (negative 2. 5) = 2. 5 Five-e
    15·2 answers
  • Which table correctly lists the three solutions for the equation 5x + y = 14?
    14·1 answer
  • Complete the table, will choose brainliest.
    8·1 answer
  • If a finite set S of nonzero vectors spans a vector space V , then some subset of S is a basis for V . True or False
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!