Question

For each of the following sets of Vectors, determine whether it is a basis in R3. If a set is a basis find the coordiantes of a = (0,1,3) in this basis.

a)
(2)(0)
(-3)(5)
(4)(6)

b)
(1)(0)(1)
(0)(1)(1)
(-1)(2)(1)

c)
(1)(0)(1)
(0)(1)(0)
(-1)(2)(1)

Answer 1

Do you know what a basis is?

Answer 2

Isn't a basis a set of one non zero vector v in the direction of l if it is for a line?

Answer 3

That's true for a line or R1, how about R3?

Answer 4

is it a set of 3 non-coplanar vectors?

Answer 5

That's right. So are the vectors from a) a basis?

Answer 6

Is it a set of three vectors?

Answer 7

Yeah it is a set of 3 vectors isn't it?

Answer 8

O wait no

Answer 9

A is only 2 sets where is B and C contain 3 sets right?

Answer 10

A is a set of two vectors and B an C are sets of three vectors.

Answer 11

Yeah that's what I mean :) What do I do after to figure out if it is a basis of a = 1,3,0...can I draw it out or is there a way to do it through an equation

Answer 12

What's your question?

Answer 13

If the set is a basis, it asks to find the coordinates of a which is (0,1,3) how do I do that?

Answer 14

Right, have you learned how to solve systems of equations? With Matrices?

Answer 15

No we haven't yet

Answer 16

Too bad, it's a lot easier if you had. Anyway let's do it for c). Finding coordinates means that you're looking for constants a, b and c, such that a*(1,0,-1)+b(0,1,2)+c(1,0,1)=(0,1,3).

Answer 17

You can see right away that b is equal to 1, because (1,0,-1) and (1,0,1) both have zeros on the second place.

Answer 18

What remains is:
a+b*0+c=0
and a*-1+2b+c=3

Answer 19

So a+c=0 and c-a=1.
So a=-c, thus c+c=1, so c=1/2 and a =-1/2

Answer 20

how do you get a+c=0 and c-a = 1? I got lost there

Answer 21

o nvm

Answer 22

you just simplified the equations since B = 0

Answer 23

I made a mistake on that second one, sorry about that.

Answer 24

Where did you make a mistake?

Answer 25

oh you mean c-a=3?

Answer 26

No it was actually fine before.

Answer 27

Sorry about this confusion.

Answer 28

a*-1+2b+c=3
-a+2+c=3
c-a=1

Answer 29

How is 2 within the equation I thought B=0?

Answer 30

b is equal to one.

Answer 31

Oh yeah! This is making sense now. What you do with a+c=o and c-a=1 after?

Answer 32

How did you get c+c=1?

Answer 33

from a+c=0 it follows that c=-a

Now I substitute c for -a in c-a=1, so you get c+c=1

Answer 34

AHHHH THIS MAKES SO MUCH SENSE YOU ARE THE BEST :) Could I ask you one more thing?

Answer 35

Sure.

Answer 36

Could you explain the idea of Linear Independence and how to figure out if a set spans linearly independent?

Answer 37

Do you know the dot product or inner product of two vectors?

Answer 38

Well she is just teaching us that but I don't think we have to apply dot product to our assignment because she isn't even finished teaching us it.

Answer 39

Two vectors are independent if there not on the same line. So (1,0,0) and (2,0,0) are dependent, while (1,2,0) and (1,0,5) are independent.

Answer 40

A vector can also be independent from a set of vectors. This is the case when that vector lies outside the plane spanned by the set.

For instance (1,0,0) is independent from the set {(0,1,0),(0,01)}.

And (0,2,3) is dependent from the set {(0,1,0),(0,0,1)}

Answer 41

So a(-1)(2) and b(1)(3) are linearly independent right?

Answer 42

What do you mean by a and b?

Answer 43

Well it asks if these sets are linearly independent which are
a)
(1) (-1) (2)
(-1) (2) (3)

b)
(-1) (1)
(2) (3)

Answer 44

A set of vectors can also be linearly independent, this is the case all the vectors are linearly independent from the rest of the vectors in the set. (So a and b are seperate questions.)

A set is dependent if you can take a combination of the vectors equal to one.

For instance if you have the set {v,w,u}, and a*v+b*w+c*u=0 for some numbers a,b c (not all zero) then it is dependent.

Answer 45

Alright thank you :) I am stuck on the Question where we have to figure out the coordinates of A. I have been trying to do B and I got to this
a+c=0
b+c=1
-a+2b+c=3
I have tried many numbers but it doesn't work? Can you guide me once again <3

Answer 46

Well, it's no wonder you can't find them, because B is not a basis. It's not an independent set.

Answer 47

so it isn't possible to do?

Answer 48

That's right.

Answer 49

Sweet Thank you! You have been awesome! Thanks for guiding me through my confusing of Linear Algebra :)

Answer 50

You're welcome. Keep at it, you'll develop a feeling for it and it'll become a lot easier.