Exercises: Gram-Schmidt Orthonormalization

Problem 1

Which of the following sets of vectors \(\Vect{v}_1,\Vect{v}_2\) are orthogonal, orthonormal or neither?

  1. \(\Vect{v}_1=(1,-2)\) and \(\Vect{v}_2=(4,2)\)

  2. \(\Vect{v}_1=(1,0)\) and \(\Vect{v}_2=(-1/\sqrt{2} , 1/\sqrt{2})\)

  3. \(\Vect{v}_1=(-1/\sqrt{5},2/\sqrt{5})\) and \(\Vect{v}_2=(2/\sqrt{5} , 1/\sqrt{5})\)

  4. \(\Vect{v}_1=(3/5,0,-4/5)\)

  5. \(\Vect{v}_1=(\sqrt{3},1,0)\), \(\Vect{v}_2=(-1,\sqrt{3} , 1)\), and \(\Vect{v}_3=(1,0,0)\).

  6. \(\Vect{v}_1=(2,-3,1,2)\), \(\Vect{v}_2=(-1,2,8,0)\), and \(\Vect{v}_3=(6,-1,1,-8)\).

Problem 2

Determine if the set of vectors \(\Set{ \Vect{v}_1,\Vect{v}_2 }\) forms a basis, an orthogonal basis, or an orthonormal basis of \(\RNrSpc{2}\), where

\[\Vect{v}_1 = (\sqrt{3},1) \quad\text{and}\quad \Vect{v}_2=(-1,\sqrt{3})\]
Problem 3

You are given the set of vectors \(\Set{ \Vect{v}_1,\Vect{v}_2 ,\Vect{v}_3 }\) with

\(\Vect{v}_1\)\(\DefEq \)\((1/3,2/3,2/3)\)
\(\Vect{v}_2\)\(\DefEq \)\((2/3,1/3,-2/3)\)
\(\Vect{v}_3\)\(\DefEq \)\((2/3,-2/3,1/3)\)

  1. Determine if these vectors form a basis, an orthogonal basis, or an orthonormal basis of \(\RNrSpc{3}\).

  2. Find the coordinates of \(\Vect{x}=(1,1,1)\) with respect to the ordered basis \(\EuScript{B}\DefEq (\Vect{v}_1,\Vect{v}_2,\Vect{v}_3)\).

Problem 4

Find an ONB of the subspace \(W\) of \(\RNrSpc{3}\) spanned by the linearly independent set of vectors \(\Set{ \Vect{w}_1, \Vect{w}_2 }\), where

\[\Vect{w}_1 = (1,0,-1) \quad\text{and}\quad \Vect{w}_2 = (1,1,0)\]
Problem 5

Find an ONB of the subspace \(W\) of \(\RNrSpc{4}\) spanned by the linearly independent set of vectors \(\EuScript{A}=(\Vect{w}_1,\Vect{w}_2,\Vect{w}_3)\) where

\[\Vect{w}_1\DefEq (1,0,1,2)\qquad \Vect{w}_2\DefEq (2,1,0,2)\qquad \Vect{w}_3\DefEq (1,-1,0,1)\]
Problem 6

You are given an orthogonal set of vectors \(B\DefEq \Set{ \Vect{v}_1,\dots ,\Vect{v}_k }\).

  1. If \(c_1,\dots ,c_k\) are nonzero numbers, show that \(C\DefEq \Set{ c_1 \Vect{v}_1,\dots , c_k \Vect{v}_k }\) is also an orthogonal set of vectors.

  2. In the statement above replace ‘orthogonal’ by ‘orthonormal’. Is this then true?