Question

linear dependent vectors in 2d space have the same slope. what do linear dependent vectors in 3d space have?

Answer 1

Two linearly dependent vectors in 3d space have the same direction, so you if took away their magnitude and made them into a unit vector, they would be equal.

Answer 2

very interesting. two linear dependent vectors can span a plane though, right? in that case the direction is different?

Answer 3

No.

Answer 4

I see

Answer 5

however,
when there are 3 vectors in 3d space

Answer 6

then a third one can be dependent to two other

Answer 7

One thing that is different though, is that in 2 space it only takes 2 linearly independent vectors to span, while in 3 spaces it takes 3 linearly independent vectors

Answer 8

yes

Answer 9

So a third vector can be linearly dependent on two linearly independent vectors. This can't happen in 2d space.

Answer 10

If the third vector is linearly dependent, then that means it is in the same plane as those two linearly independent vectors it is dependent on.

Answer 11

in 2 space, the vectors span when they have a different slope

Answer 12

In 2d space there are three types of spans:
0 vectors: point
1 vector: line
2 linearly independent vectors a plane (all of 2d space)

Answer 13

OK

Answer 14

3d space has all that as well as an infinitely large cube. which requires 3 linearly independent vectors.


However, the thing to remember is that a vector isn't linearly (in)dependent to some other vector.

It is linearly (in)dependnet to a span of vectors.

Answer 15

When you only consider two vectors, than you are comparing one vector to a span of just one vector.

Answer 16

can you compare a vector against a point?

Answer 17

That is basically comparing a vector against a span of 0 vectors.

Answer 18

ALL vectors are linearly independent to a point / span of 0 vectors.

There is no way to form a vector from a linear combination of 0 vectors!

Answer 19

It is like trying to add 0 numbers and getting a number, can't be done.

Answer 20

ok so multiplying a point doesn't really make sense

Answer 21

The only exception is maybe the \(\vec 0\) vector.

Answer 22

I'm not sure how to scalar multiply a point, anyway

Answer 23

There isn't even a point to scalar multiply, really.

Answer 24

A point is nothing, it spans nothing

Answer 25

this is illegal right?
(2, 4) * 2
=(4, 8)

Answer 26

Yes, Obviously the point ((4,8)) is not in the span of the point \((2,4)\)

Answer 27

because basically the vector from a point would be AA, or (2, 4)-(2,4)=(0, 0)

Answer 28

so no matter what the scalar is it never spans anything?

Answer 29

other than the point. no line

Answer 30

I suppose you could say the point \((2,4)\) spans \(\mathbf {\vec 0}\)

Answer 31

I dunno, just don't worry about single points or anything

Answer 32

yes, I understand the concept now. that is all that counts

Answer 33

When talking about vector subspaces and linearly independent vectors, usually we're worried about 1d +

Answer 34

I don't really worry about point spanning either

Answer 35

1 d is point ?

Answer 36

1d is line. 0d is point

Answer 37

OK, makes sense

Answer 38

+one dimension to move an additional 90° line
first no line 0d
then one line 1d
then one line and another line in 90°- 2d
then one line and another line and yet another line all 90° -3d

Answer 39

thanks for all your help

Answer 40

YEah, actually technically the vectors don't even have to be orthigonal

Answer 41

Not orthogonal yet span plane / 3d space

Answer 42

you are right. they have components and do not need to be exactly 90°

Answer 43

Anyway, I think you're smart and understand it.

Answer 44

I understand it now, thanks again