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
Sergeu [11.5K]
3 years ago
6

Let f : \mathbb{R}^{2} \to \mathbb{R}^{2} be the linear transformation defined by f(\vec{x}) = \left[\begin{array}{cc} 4 &3\

cr 2 &1 \end{array}\right] \vec{x}. Let \begin{array}{lcl} \mathcal{B} & = & \lbrace \left<1,-1\right>, \left<2,-3\right> \rbrace, \\ \mathcal{C} & = & \lbrace \left<1,1\right>, \left<-2,-1\right> \rbrace, \end{array} be two different bases for \mathbb{R}^{2}. Find the matrix \lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}} for f relative to the basis \mathcal{B} in the domain and \mathcal{C} in the codomain.
Mathematics
1 answer:
alexira [117]3 years ago
5 0

Answer:

\lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}}=\left[\begin{array}{cc} 1 &3\cr 0 &2 \end{array}\right]

Step-by-step explanation:

Remember that \lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}} is the matrix whose columns are the images under f of the vectors of the basis \mathcal{B} written in the coordinates of the basis \mathcal{C}. Then we have to do the following:

  1. Find the \mathcal{C} -- coordinates of any vector \vec{x}=\left\in\mathbb{R}^2, that is, \lbrack \vec{x} \rbrack_{\mathcal{C}}.
  2. Calculate the images under f of the vectors \vec{v_1}=\left and \vec{v_2}= \left, that is, f(\vec{v_1}) and f(\vec{v_2}).
  3. Find the \mathcal{C} -- coordinates of f(v_1) and f(v_2), that is, \lbrack f(v_1) \rbrack_{\mathcal{C}} and \lbrack f(v_2) \rbrack_{\mathcal{C}}.

For (1), note that any vector \vec{x}=\left\in\mathbb{R}^2 can be written as \\ \vec{x}=\left=(-x+2y)\left+(-x+y)\left=(-x+2y)\vec{v_1}+(-x+y)\vec{v_2}. Therefore, the \mathcal{C} -- coordinates of \vec{x} are \\ \lbrack \vec{x} \rbrack_{\mathcal{C}}=\left.

For (2), we calculate:

f(\vec{v_1}) = \left[\begin{array}{cc} 4 &3\cr 2 &1 \end{array}\right] \left[\begin{array}{c}\phantom{-}1 \cr -1 \end{array}\right]=\left[\begin{array}{c}1 \cr 1 \end{array}\right]

f(\vec{v_1}) = \left[\begin{array}{cc} 4 &3\cr 2 &1 \end{array}\right] \left[\begin{array}{c}\phantom{-}2 \cr -3 \end{array}\right]=\left[\begin{array}{c}-1 \cr \phantom{-}1 \end{array}\right]

Now we use the results obtained in steps (1) and (2) for finding \lbrack f(v_1) \rbrack_{\mathcal{C}} and \lbrack f(v_2) \rbrack_{\mathcal{C}} as requested in (3):

\lbrack f(v_1) \rbrack_{\mathcal{C}}=\left=\left

\lbrack f(v_2) \rbrack_{\mathcal{C}}\left=\left.

Therefore, the matrix \lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}} for f relative to the basis \mathcal{B} in the domain and \mathcal{C} in the codomain is given by

\lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}}=\left[\begin{array}{cc} 1 &3\cr 0 &2 \end{array}\right]

You might be interested in
If you roll a 6-sided die 6 times, what is the best prediction possible for the number of times
dlinn [17]

Answer:

I think 1

Step-by-step explanation:

Maybe not 100% $ure

3 0
2 years ago
2<br><img src="https://tex.z-dn.net/?f=%202%20%5Ctimes%202" id="TexFormula1" title=" 2 \times 2" alt=" 2 \times 2" align="absmid
NISA [10]
FOUR FOUR FOUR FOUR FOUR FOUR FOUR 2*2=FOUR
3 0
3 years ago
Read 2 more answers
I need help please and thank you
algol13

Answer:

If one angle of a triangle is 110 deg and the other two angles are both 35 deg, then the figure is an obtuse isosceles triangle.

4 0
3 years ago
Please help, I’m stuck on this one question.
Mamont248 [21]

Answer:

D the last answer

Step-by-step explanation:

As X approaches positive or negative infinity, f(x) would approach negative infinity.

6 0
2 years ago
How many kg of water is 2.2 pounds
Eddi Din [679]

0.997903 kilograms in 2.2 pounds



3 0
3 years ago
Read 2 more answers
Other questions:
  • How many of the first 10,000 positive integers contain the digit pair “43” in that order?
    8·1 answer
  • identify the coordinates four points on the line with each given slope and y -intercept slop=-1 y-intercept= 8
    7·1 answer
  • The cost per pound of chicken is $2. Each serving of chicken is about 4 oz. What will be the portion cost of the chicken?
    10·2 answers
  • Part D Complete the table to find the rule for the translation, the coordinates of triangle A′B′C′, and the coordinates of the t
    11·1 answer
  • Question 13 of 21
    14·1 answer
  • 75%
    7·2 answers
  • If u help me u r the best ✨✨✨
    11·1 answer
  • An expression is shown. 1/6 + 1/6 What is the value of the expression?
    12·2 answers
  • Can someone help me PLZ! I keep posting the same question and no ones helping me :(((((((((((( its dues in 20 mins!!
    11·2 answers
  • God bless brainiest to right answer
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!