# How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions?

I am studying and the textbook says to do this, though I am not sure where to start or how to prove a basis using orthogonality.

Part a) of the question asks to find the basis for $RowA$, $NulA$, $ColA$, and $NulA^T$. The matrix given is $A$.

$A$ = \begin{bmatrix} -1 & 2 & 4 & 9 & -11\\ 1 & -2 & 2 & 3 & -1\\ 3 & -6 & 7 & 11 & -5 \end{bmatrix} I've found a basis for $ColA$ to be {[-1,1,3],[4,2,7]}, a basis for $RowA (ColA^T)$ to be {[-1, 2, 4, 9, -11],[1, -2, 2, 3, -1]}, a basis for $NulA$ to be {[2,1,0,0,0],[1,0,-2,1,0],[-3,0,2,1]}, and a basis for $NulA^T$ to be {[-1/6, -19/6, 1]}.

The next part of the question, part b), asks to check your answers using orthogonality conditions, but it is unclear as to what these conditions are. I am hoping for some guidance.

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First, please incorporate your question into the body of the question. Second, one proves a theorem, not a basis, so it isn't clear what you mean. Please make the effort to express yourself in precise mathematically correct terms, to aid our understanding of your question. In particular, explain what you mean by "with orthogonality conditions". It may be clear to you, but you must make it clear to me. –  Gerry Myerson Nov 7 '12 at 6:32
Revised! Sorry about that. I should note that the question I'm reading isn't clear on what orthogonality conditions are, and that is where I'm confused. Thank you for the formatting advice. –  LearningPython Nov 7 '12 at 6:58
Orthonormal bases for each of the four fundamental subspaces can be read off directly from the SVD of $A$. –  littleO Nov 7 '12 at 7:10
@LittleO, how so? –  LearningPython Nov 7 '12 at 7:26
I went ahead and gave an answer below. –  littleO Nov 7 '12 at 7:28

"Check your answers using orthogonality conditions" simply means check that each basis vector for the row space is orthogonal to each basis vector for the nullspace, and each basis vector for the column space is orthogonal to each basis vector for the nullspace of the transpose.

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I'll consider the case where $A$ is an $m \times n$ matrix with real entries.

First find the full SVD of $A$: \begin{align*} A &= U \Sigma V^T \\ &= \sum_{i=1}^r \sigma_i u_i v_i^T \end{align*} where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal with nonzero diagonal entries $\sigma_1,\ldots,\sigma_r$.

A basis for the null space of $A$ is $\{v_{r+1},\ldots,v_{n}\}$. A basis for the range of $A^T$ is $\{v_1,\ldots,v_r\}$. A basis for the range of $A$ is $\{ u_1,\ldots,u_r\}$. A basis for the null space of $A^T$ is $\{u_{r+1},\ldots,u_{m}\}$.

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Ok. I'll have to figure how to extract the SVD from A. –  LearningPython Nov 7 '12 at 7:46
If your class hasn't discussed the SVD yet, perhaps they want you to use a different method. What textbook are you using, and what section are you in? –  littleO Nov 7 '12 at 7:52
Linear Algebra by David C Lay, Chapter 4 –  LearningPython Nov 7 '12 at 8:24
The edition is the 4th –  LearningPython Nov 7 '12 at 8:24
I doubt the question has anything whatever to do with the SVD. See my answer. –  Gerry Myerson Nov 7 '12 at 12:06
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