A First Course in Linear Algebra · Problem M15

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · vector-space-matrix-subsets

Problem

Consider the following sets of

3 \times 3

matrices, where the symbol

*

indicates the position of an arbitrary complex number. Determine whether or not these sets form vector spaces with the usual operations of addition and scalar multiplication for matrices. 1. All matrices of the form

\begin{bmatrix} * & * & 1 \\ * & 1 & * \\ 1 & * & * \end{bmatrix}

2. All matrices of the form

\begin{bmatrix} * & 0 & * \\ 0 & * & 0 \\ * & 0 & * \end{bmatrix}

3. All matrices of the form

\begin{bmatrix} * & 0 & 0 \\ 0 & * & 0 \\ 0 & 0 & * \end{bmatrix}

(These are the diagonal matrices.) 4. All matrices of the form

\begin{bmatrix} * & * & * \\ 0 & * & * \\ 0 & 0 & * \end{bmatrix}

(These are the upper triangular matrices.)

Related theory

S Matrix And T Matrix

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