Exercises: Orthogonal Complement

Problem 1

For each of the following sets, find its orthogonal complement inside the specified space.

  1. \(S=\Set{ 2,-4 }\) in \(\RNrSpc{1}\)

  2. \(S=\Set{ \Vect{u} }\), where \(\Vect{u} = (-3,-1)\) in \(\RNrSpc{2}\)

  3. \(S=\Set{ \Vect{0} }\), where \(\Vect{0} = (0,0)\) in \(\RNrSpc{2}\).

  4. \(S= \Set{ \Vect{u}, \Vect{v} }\), where \(\Vect{u} = (1,2)\) and \(\Vect{v} = (1,-1)\) in \(\RNrSpc{2}\).

  5. \(S=\Set{ \Vect{c} }\), where \(\Vect{c} = (-1,0,3)\) in \(\RNrSpc{3}\).

  6. \(S=\Set{ \Vect{u} , \Vect{v} }\), where \(\Vect{u} = (2,-2,1)\) and \(\Vect{v} = (1,-2,4)\) in \(\RNrSpc{3}\).

Problem 2

Find the null space of the matrix \(\Mtrx{A}\) below.

\[ \Mtrx{A} = \left[ \begin{array}{rrrr} 1 & -1 & 1 & 2 \\ 4 & 1 & 3 & 1 \\ -1 & 2 & -1 & 1 \\ 1 & 2 & 3 & 4 \end{array} \right] \]
Problem 3

Consider the linear transformation:

\[L\from \RNrSpc{4} \longrightarrow \RNrSpc{2},\quad L(x,y,z,w)=(x-y,z-w)\]

  1. Find the matrix which represents \(L\).

  2. Find the null space of \(\Mtrx{A}\).

  3. Find the kernel of \(L\).

Problem 4

If \(\Vect{s}\) and \(\Vect{t}\) are nonparallel vectors in \(\RNrSpc{3}\), explain why

\(\Set{ \Vect{s} , \Vect{t} }^{\bot}\)\(=\)\(\Set{ t(\CrssPr{ \Vect{s} }{ \Vect{t} }) \st t\in \RNr }\)
Problem 5

If \(\Vect{s}\) and \(\Vect{t}\) are parallel nonzero vectors in \(\RNrSpc{n}\), show that \(\Vect{s}^{\bot} = \Vect{t}^{\bot}\).