Let A=[ 1 1 0 1 0 0 0 0 0 0 0 0 ], 𝒳=span{

Step 1: Understand the problem and definitions. We need to find a pseudoinverse ( A^circ ) of ma

Let A=[ 1 1 0 1 0 0 0 0 ...

Let $$ A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right], \mathcal{X}=\operatorname{span}\left\{\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\right\} \text { and } \mathcal{Y}=\operatorname{span}\left\{\left[\begin{array}{l} 1 \\ 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 0 \end{array}\right]\right\} $$ Find a pseudoinverse $A^{\circ}$ of the matrix $A$ such that $\mathcal{N}_{A^{\circ}}=\mathcal{Y}$ and $\mathcal{R}_{A^{\circ}}=\mathcal{X}$.

Let
$$
A=\left[\begin{array}{lll}
1 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right], \mathcal{X}=\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]\right\} \text { and } \mathcal{Y}=\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
0 \\
1 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
1 \\
1 \\
0
\end{array}\right]\right\}
$$
Find a pseudoinverse $A^{\circ}$ of the matrix $A$ such that $\mathcal{N}_{A^{\circ}}=\mathcal{Y}$ and $\mathcal{R}_{A^{\circ}}=\mathcal{X}$.

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Key Concepts

Pseudoinverse

A pseudoinverse is a generalization of the usual matrix inverse that applies even to matrices that are not square or not of full rank. It is designed to provide a best approximate solution to systems of linear equations that may be underdetermined or overdetermined, often satisfying additional properties such as minimum norm solutions or consistency with certain projection behaviors.

Nullspace

In linear algebra, the nullspace (or kernel) of a matrix is the set of all vectors that are mapped to the zero vector by that matrix. When constructing a pseudoinverse, one may impose specific conditions on the nullspace of the inverse, ensuring that certain subspaces are annihilated or that particular solution sets are preserved in the inverse mapping.

Range

The range (or image) of a matrix is the set of all vectors that can be obtained as the output of the linear transformation represented by the matrix. In the context of pseudoinverses, specifying a desired range for the inverse ensures that the mapping of the pseudoinverse covers only a particular subspace, which can be crucial for applications where the solution is required to lie within a predetermined subspace.

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