Question

Linear algebra, Gram-Schmidt


say I have a matrix A with independent columns:

I do Gram-Schmidt to this matrix A so that I get an orthonormal matrix named Q out of it.

I understand that matrix Q will have the same collumn space as A.

But does Q have the same nullspace as A ?

And if not then, why is Q useful to us ?

Answer 1

Let me ask you a question

Answer 2

What's a basis for our familiar coordinate grid (xy plane) ?

Answer 3

isn't is [0 1] , [1 0] ?

Answer 4

it *

Answer 5

Yes, are they orthonormal ?

Answer 6

yeah

Answer 7

Why ? can't we work with some other basis that is not orthonormal ?

Answer 8

this is also called the standard basis

Answer 9

Why do we so much want our axes to orthogonal ?

Answer 10

its as simple as it can get I think

Answer 11

Yes, all the calculations become simple when you choose an orthonormal basis.
So why are you asking the usefulness of Q ?

Answer 12

because the procedure of finding Q is more complex than finding a standard basis

Answer 13

I could just find the rref instead of doing Gram - Schmidt

Answer 14

I think ..

Answer 15

Haha its an one time thing

Answer 16

Once you have a good basis for a vectorspace,
you can use it to work all the problems

Answer 17

but will it have the same nullspace as the starting A ?

Answer 18

It will right ?

Answer 19

You've said that A has independent column vectors; this means nullspace of A is {0}

Answer 20

I guess what I am really trying to pin down is : say we have a rectangular A , and I want to find its rref. Instead of row operations , I do collumn operations , or I do both row and column operations . Am I allowed to do such a thing and still arive at rref form >

Answer 21

cause Gram - Schmidt is basically col operations

Answer 22

Gram-Schmidt is about constructing an orthonormal basis from a given set of independent vectors

Answer 23

rref = reduced row echelon form

you can't mess with columns here; doing so will disturb the rowspace

Answer 24

it will disturb the row space, but will it disturb the nullspace ?

Answer 25

Exactly! messing with columns does change the nullspace

Answer 26

Consider below matrix \[A=\begin{bmatrix}1&2\\2&4 \end{bmatrix}\]

Answer 27

so for the matrix you just showed ( A ) , if I want to find Q , ill just subtruct from the second col a multiple of the first col

Answer 28

ofc its more complicated than that but essentially thats what I do

Answer 29

so we do a col operation to this matrix

Answer 30

* oh and dived by the length

Answer 31

so there is another col manipulation

Answer 32

You can't find a Q; there exists no Q, because its columns are not independent.

Answer 33

Look at that matrix again,
can you tell me its nullspace ?

Answer 34

[ -2,1 ] ?

Answer 35

Yep, next add first column to second column

Answer 36

same result

Answer 37

\[B=\begin{bmatrix}1&\color{red}{3}\\2&\color{red}{6} \end{bmatrix}\]

whats the null space of B ?

Answer 38

[ -2,1 ]

Answer 39

sure ?

Answer 40

[ -3,1 ] ?

Answer 41

Yes, so the nullspace does change with column operations

Answer 42

row operations preserve rowspace and nullspace


column operations preserve columnspace

Answer 43

i see

Answer 44

row operations preserve rowspace and nullspace, but disturb columnspace


column operations preserve columnspace, but disturb rowspace and nullspace