Question

Linear Algebra

Answer 1

I am looking at the problem lableled: A spanning set of R^3

Answer 2

It mentions that the vector equation produces the system that is in the picture. Then it goes on to say that the coefficent matrix for this system has a nonzero determinant , which ensures me that this system has a unique solution.

Answer 3

How did they go about finding the determiant?

Answer 4

meaing how can they tell its a nonzero determinat ?

Answer 5

just find the determinant like you normally would.

Answer 6

okay, so i'll use cofactor expansion

Answer 7

that would work

Answer 8

if the coeff matrix row reduces to

100
010
001

then we can be assured that the determinant is not zero

if we had a row or column of zeros produced; and tried to take the expansion along that row or column of 000s we end up with a 0 determinant

Answer 9

is there a faster way of determing whether a vector can be written as a linear combination of the vectors in a set?

Answer 10

linear algebra is NOT about speed :)

Answer 11

i understand, but this problem has like 4 subproblems, so i was thinking maybe there was a shortcut.

Answer 12

(A|b) is usually the quickest method by hand; augment A next to b to produce (I|x)

Answer 13

if we had the problem itself we could see it better :)

Answer 14

let upload it.

Answer 15

to be fair, Zarkon can see all. Me? I have to stab blindly at it till I hit a vein

Answer 16

thats sounds horrible

Answer 17

math is a beast ....

Answer 18

no that part of you stabbing yourself, math i can handle

Answer 19

okay here it comes

Answer 20

lol , not stabbing myself ... just fighting with the mathical beast, I have to find its weak point before it finds mine

Answer 21

its number 3, i just wanna do if there is a quiker way to do it

Answer 22

thats all

Answer 23

oh i got a way of doing it

Answer 24

2 2 2 copy columns 1 and 2
0 4 -12
7 5 13




2 2 2 2 2
0 4 -12 0 4
7 5 13 7 5

now multiply along the diagonals


56 -120 0
2 2 2 2 2
0 4 -12 0 4
7 5 13 7 5
103 -168 0

subtract the top numbers from the bottom

(103 -168) - (56 -120) = determinant

Answer 25

okay thanks peppy

Answer 26

youre welcome .... rango i assume lol

Answer 27

as long as the determinant aint a zero, we know the columns are linearly INdependant and since they are all R^3 compliant we can create all the vectors in R^3 with them

Answer 28

your clever

Answer 29

i have moments of lucidity :)

Answer 30

how do get the determinant of a 3 by 4 matrix

Answer 31

you can't.

Answer 32

the how do i find whether the determineat of that matrix is non zero:

Answer 33

A 3x4 matrix would imply that you have a redundant vector in its midsts.

row reduce it to find the pivot point to indicate which columns produce the most efficient span and eliminate the others to create a basis to play with

Answer 34

3 rows 4 columns ... yeah, thats right :)

Answer 35

its a coefficent matrix so it wouldnt include the 4th row, ima dummie