Linear Algebra and the C Language/a0ll

Use these two algorithms to calculate the projection of x onto R3.

* A is a subspace of R3. Find the matrix V that projects a vector x onto R3. 

            Proj(x) =  [A inv(AT A) AT] x = V x
            
* B is a basis for the orthogonal complement of AT. Find the matrix V that projects a vector x onto R3.     

      Proj(x) =  [Id - B inv(BT B) BT)] x = V x

... Wikipedia: Projection (linear algebra) 

Examples of applications in R3

• Find the transformation matrix for a projection onto R3: Proj(x) = [A inv(AT A) AT] x
• c03a.c

• Verify if AT is a basis for a Row Space by Row Reduction:
• c03b.c

• Find a basis B for the orthogonal complement of AT.
• c03c.c

• Find the transformation matrix for a projection onto R3: Proj(x) = [Id - (B inv(BT B) BT)] x
• c03d.c

Examples of applications in R4

• Find the transformation matrix for a projection onto R4: Proj(x) = [A inv(AT A) AT] x
• c04a.c

• Verify if AT is a basis for a Row Space by Row Reduction:
• c04b.c

• Find a basis B for the orthogonal complement of AT.
• c04c.c

• Find the transformation matrix for a projection onto R4: Proj(x) = [Id - (B inv(BT B) BT)] x
• c04d.c

Examples of applications in R5

• Find the transformation matrix for a projection onto R5: Proj(x) = [A inv(AT A) AT] x
• c05a.c

• Verify if AT is a basis for a Row Space by Row Reduction:
• c05b.c

• Find a basis B for the orthogonal complement of AT.
• c05c.c

• Find the transformation matrix for a projection onto R5: Proj(x) = [Id - (B inv(BT B) BT)] x
• c05d.c