Determine whether the given set S of vectors is a basis for...

Determine whether the given set $S$ of vectors is a basis for $M_{m \times n}(\mathbb{R})$. $m=3, n=2: S=\left\{\left[\begin{array}{rr}6 & -3 \\ 1 & 4 \\ 4 & -4\end{array}\right],\left[\begin{array}{rr}0 & -2 \\ 9 & 1 \\ -3 & -5\end{array}\right]\right.$ $\left.\left[\begin{array}{rr}2 & -9 \\ 1 & 1 \\ -3 & 0\end{array}\right],\left[\begin{array}{rr}1 & -5 \\ 2 & 0 \\ -4 & 0\end{array}\right],\left[\begin{array}{rr}-7 & 5 \\ 0 & -1 \\ 3 & 1\end{array}\right]\right\}$

Determine whether the given set $S$ of vectors is a basis for $M_{m \times n}(\mathbb{R})$.
$m=3, n=2: S=\left\{\left[\begin{array}{rr}6 & -3 \\ 1 & 4 \\ 4 & -4\end{array}\right],\left[\begin{array}{rr}0 & -2 \\ 9 & 1 \\ -3 & -5\end{array}\right]\right.$ $\left.\left[\begin{array}{rr}2 & -9 \\ 1 & 1 \\ -3 & 0\end{array}\right],\left[\begin{array}{rr}1 & -5 \\ 2 & 0 \\ -4 & 0\end{array}\right],\left[\begin{array}{rr}-7 & 5 \\ 0 & -1 \\ 3 & 1\end{array}\right]\right\}$

Key Concepts

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spans the entire space. This means that every vector in the space can be uniquely represented as a linear combination of the basis elements. In the context of matrices, a basis for M???(?) must consist of matrices that together provide a complete and non-redundant description of all possible matrices in that space.

Vector Space of Matrices

The set of all m by n matrices over a field, such as ?, forms a vector space where the elements are matrices and operations like matrix addition and scalar multiplication are defined. This space, denoted by M???(?), adheres to the vector space axioms, making it possible to discuss concepts such as bases, dimensions, and linear transformations within this framework.

Dimension

The dimension of a vector space is defined as the number of vectors in any basis for that space. For the space M???(?), the dimension is given by m×n, since each entry of the matrix can be independently specified. This concept is fundamental when determining whether a set of vectors (or matrices) can be a basis, as the set must contain exactly as many elements as the dimension of the space.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. For a set to be considered a basis, it must be free of redundancy; that is, removing any one of its vectors would cause it to no longer span the space. This concept is crucial when evaluating if a given set of matrices can serve as a basis.

Span

The span of a set of vectors is the set of all possible linear combinations of those vectors. A set of matrices spans a vector space like M???(?) if every matrix in the space can be expressed as a linear combination of the matrices in the set. Together with linear independence, spanning is an essential condition for a set to qualify as a basis.

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