ISTER_REG
Junior Member
- Joined
- Oct 28, 2020
- Messages
- 59
Good day,
I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the
1. row space to the null space
2. column space to the left null space
In the book of G. Strang there is this nice picture of the matrices that clarifies the orthogonality. This is what I mean:
[MATH]Ax = \begin{pmatrix}\text{row 1} \\ ... \\ \text{row m}\end{pmatrix}\begin{pmatrix} \vdots \\ x \\ \vdots \end{pmatrix} = 0[/MATH]
[MATH]A^Ty = \begin{pmatrix}\text{transposed column 1 of A} \\ ... \\ \text{transposed column n of A}\end{pmatrix}\begin{pmatrix} \vdots \\ y \\ \vdots \end{pmatrix} = 0[/MATH]
But I am more interested in the matrix proof.
I have shown the whole thing as a picture. Using this picture I try to show the matrix proof.
1. proof for the orthogonality of the row space to the null space:
[MATH] \langle\,A^Ty,x\rangle = x^T(A^Ty) = (Ax)^Ty = 0^Ty = 0[/MATH]
2. proof of the orthogonality of the column space to the lefft null space:
[MATH]\langle\, Ax, y\rangle = (Ax)^Ty = x^T(A^Ty) = x^T0 = 0[/MATH]
Are these proofs correct, so far, or have I forgotten something?
I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the
1. row space to the null space
2. column space to the left null space
In the book of G. Strang there is this nice picture of the matrices that clarifies the orthogonality. This is what I mean:
[MATH]Ax = \begin{pmatrix}\text{row 1} \\ ... \\ \text{row m}\end{pmatrix}\begin{pmatrix} \vdots \\ x \\ \vdots \end{pmatrix} = 0[/MATH]
[MATH]A^Ty = \begin{pmatrix}\text{transposed column 1 of A} \\ ... \\ \text{transposed column n of A}\end{pmatrix}\begin{pmatrix} \vdots \\ y \\ \vdots \end{pmatrix} = 0[/MATH]
But I am more interested in the matrix proof.
I have shown the whole thing as a picture. Using this picture I try to show the matrix proof.
1. proof for the orthogonality of the row space to the null space:
- The row space vectors are combinations[MATH] A^Ty[/MATH] of the rows, therefore the dot product of [MATH]A^Ty[/MATH] with any [MATH]x[/MATH] of the null space must be zero:
[MATH] \langle\,A^Ty,x\rangle = x^T(A^Ty) = (Ax)^Ty = 0^Ty = 0[/MATH]
2. proof of the orthogonality of the column space to the lefft null space:
- The column space vectors are combinations [MATH]Ax[/MATH] of the columns, therefore the dot product of [MATH]Ax[/MATH] with any [MATH]y[/MATH] of the left null space must be zero:
[MATH]\langle\, Ax, y\rangle = (Ax)^Ty = x^T(A^Ty) = x^T0 = 0[/MATH]
Are these proofs correct, so far, or have I forgotten something?