Question

Determine whether {(1,0,1,0),(1,1,0,0), (1,1,1,1), (0,0,0,1)} is a basis for R^4

Answer 1

it is

Answer 2

Yes, the book says that it is, but it does not offer an explanation for why. I'm trying to prove its linear independent and that it spans , but i dont know how to make it look like a proof

Answer 3

make a matrix ...either row reduce it or find the determinant

Answer 4

what do i want the matrix to look like? is it just reduced row with all zeros on the right side?

Answer 5

put the 4 vectors above in a matrix...put it into rref and you will get the identity matrix.

Answer 6

ok i did that and that proves linear independence, Thanks! I can see how that comes out, but now can you help me show the span?

Answer 7

you have 4 vectors that are L.I. therefore they span a subspace of dimension 4...hence they span all of R^4

Answer 8

oh by definition of a span?

Answer 9

you can also do this

let A be the above vectors in a matrix

then t be any vector in R^4

since Det(A) is not zero it is invertable

thus

there is a vector \[v\in\mathbb{R}^4\] such that

\[Av=t\]

namely \[v=A^{-1}t\]

Answer 10

what about something like -> A(1,0,0,0)+B(0,1,0,0)+C(0,0,1,0)+D(0,0,0,1). That then shows that it spans R4 after we have the linear independence proven by the matrix

Answer 11

you would have to set that equal to some arbitrary vector in R^4 and show that a solution exists

Answer 12

there is a theorem that says that

if Dim(V)=n

and S={v_1,...v_n}

is a set in L.I. vectors then S spans all of V

Answer 13

then i could just use that theorem i think.

Answer 14

it should be somewhere in your book.

Answer 15

what book are you using?

Answer 16

robert messer linear algebra gateway to mathematics
its a terrible book, its written as if its just a mere explanation to someone who already has a phd in this stuff. very little detail

Answer 17

I have many linear algebra books, but I don't have that one

Answer 18

but 3.15 states that is V is n dimensial , if v1 to vn pans v then it is l.i. and if its l.i then it spans v.

Answer 19

that's it

Answer 20

I attend MSU and they have this book only because the person that wrote it, graduated from here.

Answer 21

ic

Answer 22

I normally have no problem learning from these textbooks but i really think this one is a poor read, but i have a small quiz at 2pm today on basis and dimension.

Answer 23

ic...good luck

Answer 24

if you dont mind me asking, how often are you on this thing, you've been very helpful.

Answer 25

I on fairly often...but the time is not consistent.

Answer 26

I usually have it running in the background when I'm in my office

Answer 27

a math professor of some sort?

Answer 28

yes

Answer 29

Ph.D. Mathematical Statistics

Answer 30

thats respect, im a mathematics major, i hope to do something analytically for a career, but to have continued on to get a PhD I dont think I could handle it, thanks again! Hopefully you will be around for more of my questions. Very few people here can articulate linear algebra so well.

Answer 31

cool...and good luck again on your quiz