Department of Mathematics, K.T.H.M. College, Nashik

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Registered: 23-07-2015

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Linear Algebra

Q.(SET Nov 2011)
The dimension of the vector spaces of \(n \times n\) matrices all of whose components are 0 expect possibly the diagonal components is

A) \(n^2\)
B) \(n-1\)
C) \(n^2-1\)
D) \(n \)
Ans D
The basis of \(n \times n\) matrices all of whose components are 0 expect possibly the diagonal components ( i.e we have only diagonal elements ) are \(e_{11}, e_{22},e_{33}, ....,e_{nn} \) so that dim is n
Let A be the matrices.
\[ A_{n,n} = \begin{pmatrix} a_{1,1} & 0 & \cdots & 0 \\ 0  & a_{2,2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n,n} \end{pmatrix}
=a_{11} \begin{pmatrix} e_{11}\\ 0\\    \vdots\\ \vdots\\ \vdots\\ \vdots\\ 0 \end{pmatrix}+ a_{22} \begin{pmatrix} 0\\ e_{22}\\ 0\\    \vdots\\ \vdots\\ \vdots\\ 0 \end{pmatrix}+ \dots +a_{nn} \begin{pmatrix} 0\\ 0\\    \vdots\\ \vdots\\ \vdots\\ \vdots\\ e_{nn} \end{pmatrix}\]