( newcommand { vecs} [1] { overset { scriptstyle rightharpoonup} { mathbf {# 1}}} ) ( newcommand { vecd} [1] { overset {- ! – ! rightharpoonup} { vphantom {a} smash {# 1}}} ) ( newcommand { id} { mathrm {id}} ) ( newcommand { Span} { mathrm {span}} ) ( newcommand { kernel} { mathrm {null} ,} ) ( newcommand { range} { mathrm {range} ,} ) ( newcommand { RealPart} { mathrm {Re}} ) ( newcommand { ImaginaryPart} { mathrm {Im}} ) ( newcommand { Argument} { mathrm {Arg}} ) ( newcommand { norm} [1] { | # 1 |} ) ( newcommand { inner} [2] { langle # 1, # 2 rangle} ) ( newcommand { Span} { mathrm {span}} ) ( newcommand { id} { mathrm {id}} ) ( newcommand { Span} { mathrm {span}} ) ( newcommand { kernel } { mathrm {null} ,} ) ( newcommand { range} { mathrm {range} ,} ) ( newcommand { RealPart} { mathrm {Re}} ) ( newcommand { ImaginaryPart} { mathrm {Im}} ) ( newcommand { Argument} { mathrm {Arg}} ) ( newcommand { norm} [1] { | # 1 | } ) ( newcommand { inner} [2] { langle # 1, # 2 rangle} ) ( newcommand { Span} { mat hrm {span}} )