Question

Let A)
1 -3 -5
1 1 -2
1 -3 1
1 1 4

and b= -6
1
1
6

Answer 1

a) apply the Gram-Schmidt process to determine an orthonormal basis for R(A)

Answer 2

I have no idea

Answer 3

so b is the basis for the solution space

Answer 4

sure

Answer 5

so is b only one vector?

Answer 6

Yes

Answer 7

i am not sure, but i think we can apply an alternative form of the gram-schmidt, but if anybody else has any ideas, feel free to chime in

Answer 8

last time i did something like this i crossed v1xv2 and v1xv3 to find othogonal vectors dunno if itll work here tho;

Answer 9

dunno how we would cross 2, 4component tho .. yet :)

Answer 10

from what i have seen with the gram shmidth , B has two vectors at least

Answer 11

matrix A is already linear independant; it just aint orthogonal

Answer 12

had to look up what an orthonormal was; its an orthogonal set of vectors that are unit length

Answer 13

what is vector b? is it defined in terms of the given matrix, or is it defined in termsof the orthonromal components?

Answer 14

or is b in terms of the standard components of 1000, 0100, 0010

Answer 15

I am reading up on chapter too

Answer 16

the GS

v1 = x1; W1 = span{x1}

v2 = x2 - proj(W1)x2

-3 x2.x1
1 - ----- x1
-3 x1.x1
1

Answer 17

1 1 1 1
-3 1 -3 1
-----------
-3+1-3+1 = -4
---------- --- = -1
1+1+1+1 = 4

v2 =
-3 1 -2
1 1 2
-3 + 1 = -2
1 1 2

Answer 18

v3 = x3 - proj(W2) x3

which gets tricky

Answer 19

W2 = span{v1,v2}

Answer 20

proj(W2)x3 = the sum of the same process above but using x3 in place of x2, and dotting it to v1 and then v2

Answer 21

-5-4 1 4
1 1 1 1
---------
-5-4+1+4 = -4
---------- --- <1,1,1,1> = <-1,-1,-1,-1>
1+1+1+1 = 4


-5-4 1 4
-2 2-2 2
---------
-10-8-2+8 = -12
----------- --- <-2,2,-2,2> = <3,-3,3,-3>
4+4+4+4 = 16

-1-1-1 -1
3 -3 3 -3
---------
<2,-4,2,-4> = v3

Answer 22

\[\begin{vmatrix}1&-2&2\\1&2&-4\\1&-2&2\\1&-2&-4 \end{vmatrix}\]

would be the orthogonal basis; the orthonormal is to unit length those

Answer 23

middle bottom; not negative; just 2

Answer 24

thanks , amistre

I will study this

Answer 25

i think i made a mistake in the v3 parts

Answer 26

what i have for v3 is spose to be subtracted from x3 i think

Answer 27

-5-4 1 4 x3
-2,4,-2,4 -(badv3)
----------
-7,0,-1,8 = v3

Answer 28

much better :)

http://www.wolframalpha.com/input/?i=rref%7B%7B1%2C-2%2C-7%7D%2C%7B1%2C2%2C0%7D%2C%7B1%2C-2%2C-1%7D%2C%7B1%2C2%2C8%7D%7D

Answer 29

again, these are orthogonal basis; orthoNormal is same but unit these vectors