(a) Let S be the subspace of ℳ2 ×2 consisting of all symmetric 2 ×2 matrices. Show that S is spa

step 1 In this problem we have to verify that W is a sub space of V or not. Now we have given W is the

(a) Let S be the subspace of ℳ2 ×2 consisting of all...

(a) Let $S$ be the subspace of $\mathcal{M}_{2 \times 2}$ consisting of all symmetric $2 \times 2$ matrices. Show that $S$ is spanned by the matrices $\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$, and $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$. (b) Find a spanning set of the space of symmetric $3 \times 3$ matrices.

(a) Let $S$ be the subspace of $\mathcal{M}_{2 \times 2}$ consisting of all symmetric $2 \times 2$ matrices. Show that $S$ is spanned by the matrices $\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$, and $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$.
(b) Find a spanning set of the space of symmetric $3 \times 3$ matrices.

Key Concepts

Matrix Representation and Basis Elements

The use of specific matrices as building blocks illustrates the idea of basis elements. For instance, in the case of symmetric 2x2 or 3x3 matrices, certain matrices that have a '1' in a strategic location and '0's elsewhere capture the independent entries. These matrices form a spanning set and potentially a basis if they are linearly independent.

Symmetric Matrices

Symmetric matrices are those matrices that are equal to their own transpose. This implies that the entries above the main diagonal are mirrored by those below the main diagonal, reducing the number of independent parameters in the matrix. This concept is crucial for understanding the structure of the subspace in question.

Spanning Set

A spanning set of a vector space is a collection of vectors (or matrices, in this context) such that every element in the space can be expressed as a linear combination of these vectors. Establishing a spanning set shows that one can generate the entire space or subspace from these basic elements.

Subspace of Matrices

A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. In this problem, the set of symmetric matrices forms a subspace of the space of all matrices, allowing the application of linear algebra techniques such as finding a spanning set or basis.

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