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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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    I went ahead and gave an answer below.2012-11-07

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"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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    Ah, good point, the question is probably not asking for orthonormal bases of the four fundamental subspaces.2012-11-07