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

or5a2dosz80z

or5a2dosz80z

Answered question

2022-06-02

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document I am given the following matrix A and I need to find a nullspace of this matrix.
A = ( 2 4 12 6 7 0 0 2 3 4 3 6 17 10 7 )
I have found a row reduced form of this matrix, which is:
A = ( 1 2 0 0 23 10 0 0 1 0 13 10 0 0 0 1 22 10 )
And then I used the formula A x = 0, which gave me: A = ( 1 2 0 0 23 10 0 0 1 0 13 10 0 0 0 1 22 10 ) ( x 1 x 2 x 3 x 4 x 5 ) = ( 0 0 0 )
Hence I obtained the following system of linear equations:
{ x 1 + 2 x 2 + 23 10 x 5 = 0 x 3 + 13 10 x 5 = 0 x 4 + 22 10 x 5 = 0
How should I proceed from this point?
Thanks

Answer & Explanation

Asher Swanson

Asher Swanson

Beginner2022-06-03Added 2 answers

{ x 1 + 2 x 2 + 23 10 x 5 = 0 x 3 + 13 10 x 5 = 0 x 4 + 22 10 x 5 = 0
x 1 = 2 x 2 23 10 x 5
x 3 = 13 10 x 5
x 4 = 11 5 x 5
Therefore,basis of null space=
( 2 1 0 0 0 ) , ( 23 10 0 13 10 11 5 1 )
Aliana Kaufman

Aliana Kaufman

Beginner2022-06-04Added 13 answers

The null space of a matrix is basically a solution of the following: Ax = 0 x is a linear combination of all the independent matrices that satisfy the above equation. You have already multiplied the echelon form of A with x and have equated it to 0, so if any of the resulting equations is true, Ax = 0
We see that x5 is a variable in all the equations, so for the sake of simplicity, we take x5 = c. (a constant) Thus, x3 = -10c/13 and x4 = -10c/22 x1, however, is still a problem because of the x2 for which we have no value. So we assign x2 another value, d (also a constant) and x1 = -2d - 10c/23
The two matrices which satisfy the equation Ax = 0 come out to be (-2d 1d 0 0 0)^T and (-10c/23 0 -10c/13 -10c/22 c)^T Let's call them y and z respectively.
If Ay = 0 and Az = 0 then k(Ay) + t(Az) = 0; where k and t are constants Therefore, A (ky + tz) = 0 k * (-2d 1d 0 0 0)^T + t * (-10c/23 0 -10c/13 -10c/22 c)^T is equivalent to (if we take d and c common in their matrices) a * (-2 1 0 0 0)^T + b * (-10/23 0 -10/13 -10/22 1)^T
In short, Ax = 0 is satisfied by any linear combination of the following two matrices: (-2 1 0 0 0)^T (-10/23 0 -10/13 -10/22 1)^T
Together, they form the null space of A.
The null space of A belongs to: (-2 1 0 0 0)^T; (-10/23 0 -10/13 -10/22 1)^T

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?