Question

In ℝ3 consider the vectors 𝑢 = (1, −1,3) and 𝑣 = (2,1,0) Determine the space generated by the vectors 𝑢 and 𝑣, that is: 𝑊 =<{𝑢, 𝑣} >

likes

338 views

Answer to a math question In ℝ3 consider the vectors 𝑢 = (1, −1,3) and 𝑣 = (2,1,0) Determine the space generated by the vectors 𝑢 and 𝑣, that is: 𝑊 =<{𝑢, 𝑣} >

$Expert avatar$

Dexter

4.7

113 Answers

To determine the space generated by the vectors

\mathbf{u} = (1, -1, 3)

and

\mathbf{v} = (2, 1, 0)

\mathbb{R}^3

, we need to find all linear combinations of these vectors.

Let

\mathbf{w} = a \mathbf{u} + b \mathbf{v}

, where

and

are scalars.

Then,

\mathbf{w} = a(1, -1, 3) + b(2, 1, 0) = (a + 2b, -a + b, 3a)

\mathbf{w}=(a,-a,3a)+(2b,b,0)

\mathbf{w}=(a+2b,-a+b,3a+0)

\mathbf{w}=(a+2b,-a+b,3a)

The space generated by

\mathbf{u}

and

\mathbf{v}

is the set of all vectors

\mathbf{w} = (x, y, z)

such that there exist scalars

and

satisfying

(x, y, z) = (a + 2b, -a + b, 3a)

.

Therefore, the space

W = \text{span} \{ \mathbf{u}, \mathbf{v} \}

is the set of all linear combinations of

\mathbf{u}

and

\mathbf{v}

\mathbb{R}^3

\boxed{W = \left\{ (x, y, z) \in \mathbb{R}^3 \mid \text{ there exist } a, b \text{ such that } (x, y, z) = (a + 2b, -a + b, 3a) \right\}}

Frequently asked questions (FAQs)

What is the resultant vector of the unit vector î - 3ĵ + 2k̂ and the unit vector -2î + ĵ - 4k̂?

What is the median of the following data set: 12, 15, 17, 21, 25?

What is the measure of the third angle in a right triangle if one angle is 32 degrees?

New questions in Mathematics

8x-(5-x)

-11+29-18

The actual length of an object is 1.3 m . If the blueprint uses a scale of 1 : 12 , what is the length of the line on the drawing?

Divide 22 by 5 solve it by array and an area model

You are planning to buy a car worth $20,000. Which of the two deals described below would you choose, both with a 48-month term? (NB: estimate the monthly payment of each offer). i) the dealer offers to take 10% off the price, then lend you the balance at an annual percentage rate (APR) of 9%, monthly compounding. ii) the dealer offers to lend you $20,000 (i.e., no discount) at an APR of 3%, monthly compounding.

A National Solidarity Bond offers A 5 year bond offering a gross return of 15% Calculate the AER for this investment. (Give your answer to two decimal places, no need for the percent or € sign in your answer)

41/39 - 1/38

2x+4x=

The market for economics textbooks is represented by the following supply and demand equations: P = 5 + 2Qs P = 20 - Qd Where P is the price in £s and Qs and Qd are the quantities supplied and demanded in thousands. What is the equilibrium price?

ind the z-score for which 72% of the distribution's area lies between -z and z. -1.7417, 1.7417 -1.1538, 1.1538 -1.0803, 1.0803 -2.826, 2.826

(X+2)(x+3)=4x+18

Let X be a discrete random variable such that E(X)=3 and V(X)=5. Let 𝑌 = 2𝑋^2 − 3𝑋. Determine E(Y).

48 kg of 30% sulfuric acid in a mixture of 10% and 40% sulfuric acid arose. How many kilograms were each of the original solutions?

Translate to an equation and solve. Let x be the unknown number: What number is 52% of 81.

A buyer purchased a North Carolina home for $475,250. The seller allowed the buyer to assume his first small mortgage with a loan balance of $110,000. How much is the excise tax paid in the transaction? $951 $729.50 $950.50 $221 none of the above

Read the “Local Communities as Stakeholders: Does Amazon Really Need Tax Breaks?” example on p. 83 in Ch. 3 of Management: A Practical Introduction. In your response, discuss whether you feel that tax breaks for big companies benefit local communities. Describe ways to attract business to a region without having a negative impact on the larger community.

If the mean of the following numbers is 17, find the c value. Produce an algebraic solution. Guess and check is unacceptable. 12, 18, 21, c, 13

Determine the general solution of the equation y′+y=e−x .

Exercise The temperature T in degrees Celsius of a chemical reaction is given as a function of time t, expressed in minutes, by the function defined on ¿ by: T (t )=(20 t +10)e−0.5t. 1) What is the initial temperature? 2) Show that T' (t )=(−10 t +15)e−0 .5t. 3) Study the sign of T' (t ), then draw up the table of variations of T . We do not ask for the limit of T in +∞. 4) What is the maximum temperature reached by the reaction chemical. We will give an approximate value to within 10−2. 5) After how long does the temperature T go back down to its initial value? We will give an approximate value of this time in minutes and seconds. DM 2: study of a function Exercise The temperature T in degrees Celsius of a chemical reaction is given as a function of time t, expressed in minutes, by the function defined on ¿ by: T (t )=(20 t +10)e−0.5t. 1) What is the initial temperature? 2) Show that T' (t )=(−10 t +15)e−0.5 t. 3) Study the sign of T' (t ), then draw up the table of variations of T . We do not ask for the limit of T in +∞. 4) What is the maximum temperature reached by the reaction chemical. We will give an approximate value to within 10−2. 5) After how long does the temperature T go back down to its initial value? We will give an approximate value of this time in minutes and seconds.

The car with an irresponsible driver starts to brake when it goes through a red light. When passing the traffic light, he does so at a speed of 115 kph in the right lane. Further ahead, 70 meters from the traffic light, a child is crossing the street and falls. If the effect of the car's brakes is equivalent to a deceleration of magnitude 5.7m/s². Is the child hit by the car or not? How far from the traffic light does the car stop?

In ℝ3 consider the vectors 𝑢 = (1, −1,3) and 𝑣 = (2,1,0) Determine the space generated by the vectors 𝑢 and 𝑣, that is: 𝑊 =<{𝑢, 𝑣} >

Answer to a math question In ℝ3 consider the vectors 𝑢 = (1, −1,3) and 𝑣 = (2,1,0) Determine the space generated by the vectors 𝑢 and 𝑣, that is: 𝑊 =<{𝑢, 𝑣} >

You might be interested in