## Linear Algebra and Its Applications, Exercise 2.3.16

Exercise 2.3.16. What is the dimension of the vector space consisting of all 3 by 3 symmetric matrices? What is a basis for it?

Answer: There are nine possible entries that can be set in a 3 b 3 matrix, but if the matrix is symmetric then only six of them can be set independently, since we must have $a_{12} = a_{21}$, $a_{13} = a_{31}$, and $a_{23} = a_{32}$. Any symmetric matrix $A = \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{12}&a_{22}&a_{23} \\ a_{13}&a_{23}&a_{33} \end{bmatrix}$

can be represented as a linear combination of six linearly independent matrices as follows: $A = a_{11} \begin{bmatrix} 1&0&0 \\ 0&0&0 \\ 0&0&0 \end{bmatrix} + a_{22} \begin{bmatrix} 0&0&0 \\ 0&1&0 \\ 0&0&0 \end{bmatrix} + a_{33} \begin{bmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&1 \end{bmatrix}$ $+ a_{12} \begin{bmatrix} 0&1&0 \\ 1&0&0 \\ 0&0&0 \end{bmatrix} + a_{13} \begin{bmatrix} 0&0&1 \\ 0&0&0 \\ 1&0&0 \end{bmatrix} + a_{23} \begin{bmatrix} 0&0&0 \\ 0&0&1 \\ 0&1&0 \end{bmatrix}$

Since the above set of six linearly independent matrices spans the space of 3 by 3 symmetric matrices it is a basis for the space, and the dimension of the space is therefore six.

UPDATE: Corrected a typo in the definition of the matrix $A$. Thanks go to James Teow for finding this error.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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### 2 Responses to Linear Algebra and Its Applications, Exercise 2.3.16

1. jamesteow says:

I believe there’s a typo on the top diagram of matrix A. Row 2, Column 3 should be a23, not a13.

• hecker says:

You are correct, and I’ve updated the post to fix the error. Thanks for finding this!