Define the nullity of a matrix \(A\).

State the rank-nullity theorem for an \(m\times n\) matrix \(A\).

A matrix has \(5\) columns and rank \(3\). What is its nullity?

What is the nullity of a \(3\times3\) full-rank matrix?

Find the nullity of \(A=\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}\).

Find the nullity of the zero \(2\times4\) matrix.

If row reduction of a \(4\)-column matrix gives pivots in columns \(1\) and \(3\), what is the nullity?

Solve \(x+y=0\) as a null space and state its nullity.

Find a basis and nullity for \(A=\begin{pmatrix}1&1&1\end{pmatrix}\).

Find the nullity of \(A=\begin{pmatrix}1&2&3\\0&1&1\end{pmatrix}\).

Find a basis for the null space of \(\begin{pmatrix}1&2&3\\0&1&1\end{pmatrix}\).

Explain why the zero vector being in every null space does not mean every nullity is positive.

Find the nullity of \(A=\begin{pmatrix}1&0&1&0\\0&1&0&1\end{pmatrix}\) and give a basis for the null space.

Find the nullity of \(\begin{pmatrix}1&1&0\\0&1&1\\1&2&1\end{pmatrix}\).

For which \(t\) does \(A_t=\begin{pmatrix}1&0&1\\0&1&1\\0&0&t\end{pmatrix}\) have nullity \(0\)?

For which \(a\) does \(\begin{pmatrix}1&a\\a&1\end{pmatrix}\) have nullity \(1\)?

For which \(b\) does \(\begin{pmatrix}1&0&1\\0&1&1\b&b&1\end{pmatrix}\) have positive nullity?

A student uses the number of rows instead of columns in rank-nullity. Give a correction using a \(2\times3\) matrix of rank \(2\).

Prove that nullity equals the number of free variables in \(A\mathbf x=\mathbf0\).

Explain why nullity counts hidden input directions in a model \(\mathbf y=A\mathbf x\).