Consider an m ×n matrix A and an n ×m matrix B (with n ≠m ) such that A B=Im. (We say that A is

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Consider an m ×n matrix A and an n ×m matrix B (with n ≠m )...

Consider an $m \times n$ matrix $A$ and an $n \times m$ matrix $B$ (with $n \neq m$ ) such that $A B=I_{m}$. (We say that $A$ is a left inverse of $B$.) Are the columns of $B$ linearly independent? What about the columns of $A$ ?

Key Concepts

Linear Independence

A set of vectors is said to be linearly independent if no vector in the set can be written as a combination of the others. In the context of matrices, if the columns of a matrix are linearly independent, the matrix has full column rank. Here, the m columns of B are linearly independent, whereas the n columns of A (with n > m) cannot all be independent.

Dimensionality and Rectangular Matrices

Rectangular matrices have differing numbers of rows and columns, which affects their ability to have linearly independent sets of columns or rows. In a matrix where the number of columns exceeds the number of rows (n > m), it is impossible for all columns to be linearly independent. Conversely, in a matrix with fewer columns than rows, the columns can be linearly independent if the matrix has full column rank. This idea is critical in understanding why in our case, the columns of B (with m columns) are independent, while those of A (with n > m columns) are not.

Left Inverse

A matrix A is called a left inverse of another matrix B if the product A·B equals the appropriate identity matrix. This means that A 'undoes' the action of B from the left. In the context of rectangular matrices, this implies strong restrictions on the ranks and dimensions of the matrices involved.

Matrix Rank

The rank of a matrix is the maximum number of its linearly independent columns (or rows). When a product of matrices equals an identity matrix, it forces both matrices to have full rank relative to the dimensions of that identity. For example, if A·B equals the m×m identity, then both A (an m×n matrix) and B (an n×m matrix) must have rank m.

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