Basis for ROW(A), COL(A) and NUL of a square matrix

In summary, the conversation discusses solving equations for the null space, with examples using vector form and constant vectors as a basis. The conversation also mentions frustration and a previous deletion.
  • #1
Susanne217
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deleted

No one answered
 
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  • #2
Looks fine up until the null space. Ax=0 yields the equations

[tex]x_1+(1+i)x_2+x_4 = 0[/tex]
[tex]x_3+5ix_4 = 0[/tex]

You have four unknowns and only two equations, so you can solve for two of the variables in terms of the other two.

[tex]x_1 = -(1+i)x_2-x_4[/tex]
[tex]x_3 = -5ix_4[/tex]

In vector form, this would be

[tex]\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix} = x_2\begin{pmatrix}{-1-i\\1\\0\\0}\end{pmatrix} + x_4\begin{pmatrix}-1\\0\\-5i\\1\end{pmatrix}[/tex]

The two constant vectors are a basis of the null space.
 
  • #3
Sorry I deleted it. I thought no would answer and got frustrated. But thanks.
 

1. What is the basis for the row space of a square matrix?

The basis for the row space of a square matrix is the set of linearly independent rows that span the entire row space. These rows are the minimum number of vectors needed to represent all other rows in the matrix.

2. How is the basis for the column space of a square matrix determined?

The basis for the column space of a square matrix is determined by finding the linearly independent columns that span the entire column space. These columns are the minimum number of vectors needed to represent all other columns in the matrix.

3. What is the relationship between the basis for the row space and column space of a square matrix?

The basis for the row space and column space of a square matrix are related through the concept of duality. This means that the basis for the row space is the transpose of the basis for the column space, and vice versa.

4. How does the basis for the null space of a square matrix relate to its row and column spaces?

The basis for the null space of a square matrix is comprised of all the vectors that are orthogonal (or perpendicular) to the row space and column space. This means that the basis for the null space is the set of solutions to the homogeneous equation Ax = 0, where A is the square matrix.

5. Can the basis for the row space, column space, and null space of a square matrix be empty?

Yes, it is possible for the basis of any of these spaces to be empty. This would occur when the matrix has all zero rows or columns, meaning that there are no linearly independent vectors that can span the respective space. In this case, the dimension of the space would be 0.

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