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In linear algebra, the rank of a matrix is the dimension of the vector space generated by its columns. This is equivalent to the maximum number of linearly independent column vectors in the matrix. Example: For matrix $A = \begin{pmatrix}1 & 2\\ 3 & 6\end{pmatrix}$, rank(A) = 1 since the second column is a multiple of the first.

The Rank-Nullity Theorem states that for any finite-dimensional vector space, the rank of a matrix plus its nullity equals the number of columns of the matrix (the dimension of the vector space $\mathbb{R}^n$). Example: If a matrix has a rank of 3 and 5 columns, then nullity = 5 - 3 = 2.

The rank of a matrix can be determined by converting the matrix to its row-echelon or reduced row-echelon form through Gaussian elimination. The number of non-zero rows after this transformation is the rank. Example: $A = \begin{pmatrix}1 & 2\\ 2 & 4\end{pmatrix}$, after row reduction, $R = \begin{pmatrix}1 & 2\\ 0 & 0\end{pmatrix}$, rank(A) = 1.

The column space of a matrix is the span of its column vectors. The dimension of the column space is the rank of the matrix. Example: For matrix $A = \begin{pmatrix}1 & 3\\ 2 & 6\\ 3 & 9\end{pmatrix}$, the column space is all linear combinations of $\begin{pmatrix}1\\ 2\\ 3\end{pmatrix}$, and rank(A) = 1.

Nullity of a matrix refers to the dimension of the kernel (null space) of the matrix, which is the set of all vectors that map to the zero vector when multiplied by the matrix. Example: If $A = \begin{pmatrix}1 & 2 & 3\\ 4 & 5 & 6\end{pmatrix}$, nullity(A) = 1 since there's one free variable when solving $Ax = 0$.

A matrix has an inverse if and only if its nullity is 0. If there are no non-trivial solutions to $Ax = 0$, the matrix is invertible (non-singular). Example: Nullity(A) = 0 implies A is an invertible matrix.

The null space of a matrix is the set of all vectors that map to zero. A basis for this space can be found by solving $Ax = 0$. The number of vectors in the basis is the nullity of the matrix. Example: For $A = \begin{pmatrix}1 & 2 & -1\\ 2 & 4 & -2\end{pmatrix}$, a basis for the null space is $\begin{pmatrix}-2\\ 1\\ 0\end{pmatrix}$, and nullity(A) = 1.

The rank of a product of two matrices is less than or equal to the rank of each of the matrices. For matrices $A$ and $B$, rank(AB) ≤ min{rank(A), rank(B)}. Example: If rank(A) = 2 and rank(B) = 3, then rank(AB) ≤ 2.

The row space of a matrix is the span of its row vectors. The dimension of the row space is also the rank of the matrix. Example: For matrix $A = \begin{pmatrix}1 & 2\\ 0 & 0\end{pmatrix}$, the row space is spanned by $\begin{pmatrix}1 & 2\end{pmatrix}$, and rank(A) = 1.

An invertible matrix has full rank, meaning the rank is equal to the number of rows (and columns) of the matrix. Example: Any 3x3 invertible matrix will have a rank of 3.

In the context of linear transformations, the dimension of the kernel (nullity) plus the dimension of the image (rank) equals the dimension of the domain, $V$. Example: A linear map $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ with rank 2 has a nullity of 1, since $2 + 1 = 3$.

Elementary row and column operations do not affect the rank of a matrix. These operations include row swapping, row addition, and row multiplication. Example: Regardless of row operations, the rank of $A = \begin{pmatrix}1 & 4\\ 2 & 8\end{pmatrix}$ remains 1.

If $B$ is a subspace of $A$, then rank($B$) ≤ rank($A$). The dimension of a subspace cannot exceed the dimension of the space it's contained within. Example: If $A$ is a matrix with rank 4, any subspace of the column space of $A$ has a rank of 4 or less.

The rank of a matrix is equal to the rank of its transpose. For matrix $A$, rank(A) = rank($A^T$). Example: If $A = \begin{pmatrix}1 & 1\\ 2 & 2\end{pmatrix}$, both A and $A^T$ have a rank of 1.