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
natita [175]
3 years ago
6

Determine whether the set of all linear combinations of the following set of vector in R^3 is a line or a plane or all of R^3.a.

{(-2,5,-3), (6, -15,9),(-10, 25, -15)} b. {(1,2,0), (1,1,1),(4,5,3)} c. {(0,0,3), (0,1,2), (1,1,0)}
Mathematics
1 answer:
Temka [501]3 years ago
4 0

Answer:

a. Line

b. Plane

c. All of R^3

Step-by-step explanation:

In order to answer this question, we need to study the linear independence between the vectors :

1 - A set of three linearly independent vectors in R^3 generates R^3.

2 - A set of two linearly independent vectors in R^3 generates a plane.

3 - A set of one vector in R^3 generates a line.

The next step to answer this question is to analyze the independence between the vectors of each set. We can do this by putting the vectors into the row of a R^(3x3) matrix. Then, by working out with the matrix we will find how many linearly independent vectors the set has :

a. Let's put the vectors into the rows of a matrix :

\left[\begin{array}{ccc}-2&5&-3\\6&-15&9\\-10&25&-15\end{array}\right] ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix  ⇒

\left[\begin{array}{ccc}-2&5&-3\\0&0&0\\0&0&0\end{array}\right]

We find that the second vector is a linear combination from the first and the third one (in fact, the second vector is the first vector multiply by -3).

We also find that the third vector is a linear combination from the first and the second one (in fact, the third vector is the first vector multiply by 5).

At the end, we only have one vector in R^3 ⇒ The set of all linear combinations of the set a. is a line in R^3.

b. Again, let's put the vectors into the rows of a matrix :

\left[\begin{array}{ccc}1&2&0\\1&1&1\\4&5&3\end{array}\right] ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒

\left[\begin{array}{ccc}1&1&1\\0&1&-1\\0&0&0\end{array}\right]

We find that there are only two linearly independent vectors in the set so the set of all linear combinations of the set b. is a plane (in fact, the third vector is equivalent to the first vector plus three times the second vector).

c. Finally :

\left[\begin{array}{ccc}0&0&3\\0&1&2\\1&1&0\end{array}\right] ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒

\left[\begin{array}{ccc}1&1&0\\0&1&2\\0&0&3\end{array}\right]

The set is linearly independent so the set of all linear combination of the set c. is all of R^3.

You might be interested in
Given just an angle, X, calculate the sine and cosine of the angle. No sides are provided. Show the formula.
zhannawk [14.2K]
This is the formula

6 0
3 years ago
There are 15 red, 10 green and 9 white balls in the dark room. Jack wants to take a minimal number of balls such that at least 8
agasfer [191]
<h3>Answer: 24</h3>

=================================================

Explanation:

If Jack were to get really lucky and draw either

* 8 red balls in a row

* 8 green balls in a row, or,

* 8 white balls in a  row

then the answer would be 8. However, this event of drawing the same color ball 8 times in a row is fairly unlikely. What is more likely is that there will be multiple colors involved (because we have roughly the same number for each color mors or less). So consider the scenario in the next section below.

----------

If Jack were to draw a red ball first, then a green ball second, and then a white ball third, then so far has selected 3 balls from the drawer. None of the colors match up. This is leading to the worst case scenario in terms of the number of balls to select. In other words, Jack is really unlucky to not get any colors match up so far. When aiming for a guarantee like this, it is wise to think of the worst case scenario.

If we repeat the pattern (red, green, white) then so far we have 2 balls of each color. In total we have selected 6 overall. Note how 3*2 = 6.

Repeat this a third time, then we'll select 9 balls total (3*3 = 9)

The fourth iteration of this pattern has 3*4 = 12 balls overall picked out, and so on.

If we continue the pattern, then we'll see that we will need to select 3*8 = 24 balls to guarantee that we have at least 8 of the same color (eg: 8 red balls). Chances are that we'll have 8 of the same color before we hit the 24 ball mark, but we wont have a 100% guarantee of such. Reaching 24 balls is the only way to guarantee the claim is true.

So to summarize: I pictured the worst case scenario (red,green,white) and extended out the pattern so that it led to 24 as the final answer.

4 0
3 years ago
Read 2 more answers
Lines a and b are perpendicular. The slope of line a is −2. What is the slope of line b?
fiasKO [112]

b= 1/2

perpendicular slopes are those that are opposite signs and reciprocal of the original given slope

3 0
3 years ago
Read 2 more answers
3 out of every 5 picks are orange. if 12 picks are orange, how many picks in all?
katen-ka-za [31]
20 picks 
is the answer

4 0
3 years ago
A comedian knows eleven jokes. One joke is​ old, one joke is​ new, and the other jokes are somewhere between. If the order in wh
Nadya [2.5K]

Answer:

IDk some kind of riddle

8 0
3 years ago
Read 2 more answers
Other questions:
  • Olive’s solar powered scooter travels at a rate of 30 miles per hour. What equation can she use to calculate her distance with r
    9·2 answers
  • TVS are on sale at best buy you decide to buy 3 tvs over the phone and that the total will be $2697.00 but arent sure of each tv
    15·1 answer
  • Two consecutive integers add to 51. what are the integers?
    5·2 answers
  • Charmaine is on the swim team. Each day she swims 950m . How far does she swim in 5 days?
    7·2 answers
  • A baker has 24 sesame bagels and 36 plain bagels to put into boxes each box must have the same number of each type of bagel what
    9·1 answer
  • Two triangles with the same corresponding side lengths will be congruent is known as the ?
    11·2 answers
  • What measurement do you use for surface area
    8·1 answer
  • If a van traveled 200 miles in 5 hours how far would it travel in 8 hours
    7·2 answers
  • 6. A total of 1,022 guests are attending a reception. If eachtable can seat
    7·2 answers
  • Help me please as soon as possible
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!