Write Bold x as the sum of two vectors, one in Span StartSet Bold u 1 comma Bold u 2 comma Bold u 3 EndSet and one in Span StartSet Bold u 4 EndSet. Assume that StartSet Bold u 1 comma ... comma Bold u 4 EndSet is an orthogonal basis for set of real numbers R Superscript 4. Bold u 1equalsStart 4 By 1 Table 1st Row 1st Column 0 2nd Row 1st Column 1 3rd Row 1st Column negative 7 4st Row 1st Column negative 1 EndTable , Bold u 2equalsStart 4 By 1 Table 1st Row 1st Column 6 2nd Row 1st Column 8 3rd Row 1st Column 1 4st Row 1st Column 1 EndTable , Bold u 3equalsStart 4 By 1 Table 1st Row 1st Column 1 2nd Row 1st Column 0 3rd Row 1st Column 1 4st Row 1st Column negative 7 EndTable , Bold u 4equalsStart 4 By 1 Table 1st Row 1st Column 8 2nd Row 1st Column negative 6 3rd Row 1st Column negative 1 4st Row 1st Column 1 EndTable , Bold xequalsStart 4 By 1 Table 1st Row 1st Column 12 2nd Row 1st Column negative 7 3rd Row 1st Column 2 4st Row 1st Column 0 EndTable

Verify that

StartSet Bold u 1 comma Bold u 2 EndSetu1,u2

is an orthogonal set, and then find the orthogonal projection of y onto

SpanStartSet Bold u 1 comma Bold u 2 EndSetu1,u2.

yequals=left bracket Start 3 By 1 Matrix 1st Row 1st Column 4 2nd Row 1st Column 2 3rd Row 1st Column negative 1 EndMatrix right bracket

4
2
−1
, Bold u 1u1equals=left bracket Start 3 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column 4 3rd Row 1st Column 0 EndMatrix right bracket

3
4
0
, Bold u 2u2equals=left bracket Start 3 By 1 Matrix 1st Row 1st Column negative 4 2nd Row 1st Column 3 3rd Row 1st Column 0 EndMatrix right bracket

−4
3
0