I am given the following matrix A and I need to find a nullspace of...

Reed Eaton

Reed Eaton

Answered

2022-06-25

I am given the following matrix A and I need to find a nullspace of this matrix.
A = ( 2 1 4 1 1 1 1 1 1 0 3 2 3 2 5 0 )
I have found a row reduced form of this matrix, which is:
A = ( 1 0 3 2 0 1 2 3 0 0 0 0 0 0 0 0 )
And then I used the formula A x = 0, which gave me:
A x = ( 1 0 3 2 0 1 2 3 0 0 0 0 0 0 0 0 ) ( x 1 x 2 x 3 x 4 ) = ( 0 0 0 0 )
Hence I obtained the following system of linear equations:
{ x 1 + 3 x 3 2 x 4 = 0 x 2 2 x 3 + 3 x 4 = 0
So I just said that x 3 = α, x 4 = β and the nullspace is:
n u l l s p a c e ( A ) = { 2 β 3 α , 2 α 3 β , α , β )   |   α , β R }
Is my thinking correct?

Answer & Explanation

Zayden Wiley

Zayden Wiley

Expert

2022-06-26Added 21 answers

Since x 1 = 2 x 4 3 x 3 and x 2 = 2 x 3 3 x 4
if ( x 1 , x 2 , x 3 , x 4 ) nullspace(A):
( x 1 , x 2 , x 3 , x 4 ) = ( 2 x 4 3 x 3 , 2 x 3 3 x 4 , x 3 , x 4 ) = x 3 ( 3 , 2 , 1 , 0 ) + x 4 ( 2 , 3 , 0 , 1 )
So Nullspace ( A ) = ( 3 , 2 , 1 , 0 ) , ( 2 , 3 , 0 , 1 ) .

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