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

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

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

Key Concepts

Spanning

Spanning a vector space means that any vector in the space can be constructed as a linear combination of the given set of vectors. For a set to form a basis, it must span the entire space. When dealing with M???(?), a spanning set must allow the representation of any 2×2 matrix. If the set does not span the space, then it cannot serve as a basis regardless of its linear independence.

Linear Independence

Linear independence refers to a set of vectors where no vector can be written as a linear combination of the others. In the context of a basis, it is essential that all vectors in the set are linearly independent to avoid redundancy. If a set contains more vectors than the dimension of the space—in this case, five matrices in a four-dimensional space—the set must be linearly dependent.

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spanning for the space. This means that every vector in the space can be uniquely expressed as a linear combination of the basis vectors. In the context of matrix spaces, determining whether a given set of matrices forms a basis requires checking these two key properties.

Vector Space

A vector space is a set equipped with two operations—vector addition and scalar multiplication—that satisfy certain axioms, such as associativity, commutativity of addition, and distributivity. In the given problem, the set of m×n matrices over the real numbers, M???(?), forms a vector space, and understanding the properties of a vector space is fundamental to analyzing bases and dimension.

Dimension

The dimension of a vector space is defined as the number of vectors in any basis for that space. For the space of m×n matrices, the dimension is m×n because each entry in the matrix is independent of the others. Specifically, for M???(?), the dimension is 4, which implies that any basis for this space must consist of exactly four linearly independent matrices.

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