Question

Hi I need some help remembering the definition and conditions of a basis:
In which way can the following implications go?
1) => 2), 2) => 1) or both?
1) A basis for a subspace or any space has to satisfy the following 2 conditions:
a) be linearly independent, b) span the subspace;
2) The "Basis theorem 15" in David C. Lay's 3rd edition of Linear Algebra and its applications says:
Let H be a p-dimensional subspace of R^n. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H that spans H is automatically a basis for H.

Answer 1

I'm confused because:
2) seems to suggest that condition a) implies condition b) for whichever it satifies, the other must also be true. but in 1) it seems to suggest that both a) and b) need to be satisfied before we can even conclude it being a basis to begin with.

Answer 2

ya what about it

Answer 3

its say if u have p dimensional subspace, then having exactly p elements, whose lin combination is able to spam all of H, means they must be lin independant

Answer 4

it's both

Answer 5

so does 2) mean p elements will always span all of p dimension so only L.I. is enough? and having p elements is also equivalent of saying it is L.I. so it just has to span in order to be basis? 2) just seems so confusing sorry

Answer 6

a) be linearly independent, b) span the subspace;
p dim space
u can only have p lin independant vectors at most

Answer 7

actually hmm

Answer 8

nvm this isnt really right

Answer 9

1 doesnt really have to imply 2

Answer 10

because for example
if u have z=0 plane in r^3, your vectors will be dimension <x,y,0> which is p3

Answer 11

but ud only need 2 independant vectors to define your whole basis

Answer 12

and 2 independant vectors in the xy plane

Answer 13

so 1 doesnt imply 2

Answer 14

but 2 implies 1

Answer 15

I still don't get what i wrote in my first reply to my own post sorry.

Answer 16

meh dont think too hard about it

Answer 17

theyre trying to make a big deal out of something trivial

Answer 18

there are 3 things u need to know really

Answer 19

"u can only have p lin independant vectors at most"
but you can still have less and they still form a basis or not?

Answer 20

1) A vector space contains the 0 vector
2) k*U, a scalar times any vector in the vector space is still in the vector space
3)k1*U+k2*V , any linear combinations of vectors in a vector space stay in the vector space

Answer 21

yes that is right

Answer 22

like for example u can be in r^1000000000000, but still have only the plane in xy

Answer 23

<x,y,0,0,0,0,...,0>

Answer 24

this is a dimension 100 vector however, only 2 lin independant vectors are needed to define this plane

Answer 25

oh oops i read the question wrong this is what they are saying

Answer 26

they say u just need p elements but the condition is your p element vectors are lin independant

Answer 27

so ya its still both!

Answer 28

1 implies 2 and 2 implies 1

Answer 29

ok i get that part, but the "Also, any set of p elements of H that spans H is automatically a basis for H. "

does that statement mean the Linear independence is assumed, or the previous sentence's L.I. is already taken into account?

Answer 30

no not true

Answer 31

they have to be lin independant

Answer 32

u can have 10 p element vectors and still span only r3

Answer 33

where as u can span that r^3 with only 3 independant p element vectors

Answer 34

yea that's what I thought too, thanks.

Answer 35

pls pay dan815

Answer 36

that is my QH accnt :)

Answer 37

byee

Answer 38

Is dan815 going to make a post here for me to 'best response' it? sorry I'm such a newbie here

Answer 39

oh u hae to click rate a quahlified helper orange button

Answer 40

the best response thing u can give u anyone

Answer 41

I clicked it, nothing shows up

Answer 42

ah okay dont worry about it, i guess its because no QH responded

Answer 43

xD well cheers!

Answer 44

thanks!

Answer 45

k