Compute the orthogonal projection of left bracket Start 2 By 1 Matrix 1st Row 1st Column negative 4 2nd Row 1st Column 5 EndMatrix right bracket onto the line through left bracket Start 2 By 1 Matrix 1st Row 1st Column negative 1 2nd Row 1st Column 1 EndMatrix right bracket and the origin. The orthogonal projection is left bracket Start 2 By 1 Matrix 1st Row 1st Column nothing 2nd Row 1st Column nothing EndMatrix right bracket . Bold y equals=left bracket Start 2 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column 6 EndMatrix right bracket 3 6 and Bold uuequals=left bracket Start 2 By 1 Matrix 1st Row 1st Column 6 2nd Row 1st Column negative 6 EndMatrix right bracket 6 −6.
Write Bold y as the sum of two orthogonal vectors, one in Span Start Set Bold u End Set Span {u} and one orthogonal to Bold u.