Problem 22 - Differential Equations and Linear Algebra

Question

Let \(S\) be the subspace of \(M_{3}(\mathbb{R})\) consisting of all \(3 \times 3\) symmetric matrices. Find a set of vectors that spans \(S .\)

Step-by-Step Solution

Answer

A set of vectors that spans the subspace \(S\) of all \(3 \times 3\) symmetric matrices is: \[\left\lbrace \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}, \begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix} \right\rbrace\]

Step 1: Find the general form of a symmetric matrix

Recall that a matrix is symmetric if it is equal to its transpose. For a \(3 \times 3\) matrix \(A=\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{pmatrix}\), we have \(A = A^T\) if \(a_{21}=a_{12}\), \(a_{31}=a_{13}\), and \(a_{32}=a_{23}\). Therefore, the general form of a \(3 \times 3\) symmetric matrix is: \[A = \begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33}\end{pmatrix}\]

Step 2: Express the symmetric matrix as a linear combination of vectors

We can now express the symmetric matrix \(A\) as a linear combination of six vectors, one for each of the unique elements of \(A\): \[A = a_{11}\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} + a_{22}\begin{pmatrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix} + a_{33}\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}\] \[+ a_{12}\begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} + a_{13}\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{pmatrix} + a_{23}\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}\]

Step 3: Identify the set of vectors that spans the subspace S

From the expression in Step 2, we can see that any \(3 \times 3\) symmetric matrix can be written as a linear combination of the following set of vectors: \[\left\lbrace \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}, \begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix} \right\rbrace\] This set of vectors spans the subspace \(S\) of all \(3 \times 3\) symmetric matrices.

Key Concepts

Spanning Set of Vectors

Understanding the concept of a spanning set of vectors is fundamental in linear algebra, as it pertains to the idea of coverage within a vector space. A set of vectors spans a vector space if every other vector in the space can be expressed as a linear combination of this set. In simpler terms, if you think of the vector space as a room, then a spanning set of vectors would be enough to cover every inch of the floor.

For example, in the exercise provided, we are asked to identify a set of vectors that spans the subspace of symmetric matrices. This means we need to find a collection of symmetric matrices such that any other symmetric matrix can be written as a sum of these matrices, each multiplied by appropriate coefficients. The solution has successfully provided such a set for 3x3 symmetric matrices. It’s essential to grasp that having this set allows us to describe every possible symmetric matrix of that size, showcasing the power of a spanning set.

Linear Algebra

Linear algebra is like the grammar of mathematics; it is the study of vectors, vector spaces, linear mappings, and systems of linear equations. It is a language that underlies many areas of mathematics and is widely used in applications such as science, engineering, computer graphics, and social sciences.

The concept of a symmetric matrix and the ability to express it as a combination of a spanning set are topics deeply embedded in linear algebra. The exercises related to these concepts not only help in understanding the structure and properties of vector spaces but also enable solutions to practical problems like optimizing systems or understanding rotations and transformations in space.

Matrix Transpose

The transpose of a matrix, denoted by the superscript 'T', is a new matrix created by swapping the rows and columns of the original. For instance, if you flip a chessboard on its side, the columns become rows and vice versa - that's akin to transposing a matrix.

Regarding symmetric matrices, which are the focus of our exercise, they hold a mirror in the middle; their transpose is equal to the original matrix. This means that if you flip the matrix along its main diagonal, you will end up with the same matrix. Mathematically, if matrix A is symmetric, then A equals its transpose, that is, \( A = A^T \).

Linear Combination

Imagine you're making a new paint color by mixing other colors. Each color is a 'vector,' and the amount of each paint you add is like the 'scalar.' Mixing these creates your new color, just like a linear combination combines vectors with scalars to create new vectors.

In our exercise, we use this technique to show that any symmetric 3x3 matrix can be expressed as a linear combination of a specific set of six matrices. By adjusting the amounts of each 'ingredient' matrix, we can produce every possible symmetric matrix in the subspace. This is a perfect illustration of linear combinations in action and underscores their vital role in linear algebra.