Linear Algebra (Symmetric Matrix)

[MoBaT]

A 3x3 symmetric matrix has a null space of dimension one containing the vector (1,1,1). Find the bases and dimensions of the column space, row space, and left null space.

I understand how to get the Dim of Col(A), Row(A), and Nul(A^T) but how do i get the bases with just knowing the dimension of one vector? How should I approach this?

 

[chiro]

Hey MoBaT.

Can you pick any linearly independent basis?

 

[MoBaT]

Hey MoBaT.

Can you pick any linearly independent basis?

Dosen't say anything against it. I know what the answer is becuase he gave it to us. It was like:

Dim of Col(A) = 2, dim Row(A) = 2, Dim Nul(A^T) = 1
Basis Row(A) = {[-1 1 0], [-1 - 1]}. Because A is symmetric, Col(A) = Row(A) and Nul(A^T) = Nul(A).

I understand everything after the Basis row(A) but do not understand how he got that Row(A)

 

[MoBaT]

Found out how to do it. Pretty much fill in the rest of the information with the identity matrix.

 

[blue_raver22]

To find the bases and dimensions of the column space and row space, we can use the fact that the null space of a symmetric matrix is perpendicular to its column space and row space. This means that the vector (1,1,1) is perpendicular to all the columns and rows of the symmetric matrix.

For the column space, this means that the column space is spanned by the two vectors that are perpendicular to (1,1,1). One possible basis for the column space could be (1,0,-1) and (0,1,-1). Therefore, the dimension of the column space is 2.

For the row space, we can use the same approach and find that the row space is also spanned by (1,0,-1) and (0,1,-1). Therefore, the dimension of the row space is also 2.

For the left null space, we can use the fact that the left null space of a matrix A is the null space of its transpose, A^T. Since we know that the null space of A^T is spanned by (1,1,1), the left null space of A is spanned by the vector (1,1,1). Therefore, the dimension of the left null space is 1 and the basis is (1,1,1).

Overall, the bases and dimensions of the column space, row space, and left null space for a 3x3 symmetric matrix with a null space of dimension one containing the vector (1,1,1) are:

- Column space: dimension 2, basis: (1,0,-1), (0,1,-1)
- Row space: dimension 2, basis: (1,0,-1), (0,1,-1)
- Left null space: dimension 1, basis: (1,1,1)