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Unformatted text preview: m ) and c ∈ R . Solution. By de±nition, [ T ] is the unique m × n matrix such that T ( x ) = [ T ] x for all x ∈ R n . Since matrix algebra gives ([ S ] + [ T ]) x = [ S ] x + [ T ] x = S ( x ) + T ( x ) = ( S + T )( x ) , it follows that [ S + T ] = [ S ] + [ T ]. Similarly, since ( c [ T ]) x = c ([ T ] x ) = c ( T ( x )) = ( cT )( x ) , we have [ cT ] = c [ T ]. (Direct calculations using the formula [ T ] ij = ± e i , T ( e j ) ² are also possible.) Date : January 12, 2012. S. Kaliszewski, Department of Mathematics, Arizona State University. 1...
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