Exercises: Linear Independence

Problem 1

Determine whether the following subsets of \(\RNrSpc{2}\) are linearly independent.

  1. \(S=\Set{ (0,0) }\)

  2. \(S=\Set{ (9,9) }\)

  3. \(S=\Set{ (1,1),(9,9) }\)

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

  5. \(S=\Set{(1,1), (-1,1),(3,-4) }\)

Problem 2

Determine whether the following subsets of \(\RNrSpc{3}\) are linearly independent.

  1. \(S = \Set{ (1,3,-2),(0,-1,2) }\)

  2. \(S = \Set{ (1,3,-2),(0,-1,2),(0,0,0) }\)

  3. \(S = \Set{ (1,3,-2),(0,-1,2),(-2,1,1) }\)

Problem 3

Determine if the following vectors from \(\RNrSpc{4}\) form a linearly independent set:

\(\Vect{w}_1\)\(\DefEq \)\((1,-1,2,5)\)
\(\Vect{w}_2\)\(\DefEq \)\((3,1,4,2)\)
\(\Vect{w}_3\)\(\DefEq \)\((1,1,0,0)\)
\(\Vect{w}_4\)\(\DefEq \)\((5,1,6,7)\)
Problem 4

For each of the problems below determine if it is true or false.

  1. Any two distinct vectors in \(\RNrSpc{2}\) are linearly independent.

  2. Any two non parallel vectors in \(\RNrSpc{3}\) are linearly independent.

  3. If \(\Vect{x},\Vect{y},\Vect{z}\) are three unit vectors in \(\RNrSpc{n}\) which are perpendicular, then these vectors are linearly independent.

  4. If \(\Vect{x}=(1,0,1)\) and \(\Vect{y}=(1,1,0)\), then \(S \DefEq \Set{ \Vect{x},\Vect{y}, \CrssPr{ \Vect{x} }{ \Vect{y} } }\) is linearly independent.

  5. If \(\Vect{x},\Vect{y}\) are nonzero vectors in \(\RNrSpc{3}\), then \(S\DefEq \Set{ \Vect{x},\Vect{y}, \CrssPr{ \Vect{x} }{ \Vect{y} } }\) is linearly independent.

  6. If \(\Vect{x},\Vect{y}\) are nonparallel vectors in \(\RNrSpc{3}\), then \(S\DefEq \Set{ \Vect{x},\Vect{y}, \CrssPr{ \Vect{x} }{ \Vect{y} } }\) is linearly independent.

Problem 5

Given the vector \(\Vect{x}\) in \(\RNrSpc{2}\), find a vector \(\Vect{y}=(u,v)\) such that \(S\DefEq \Set{ \Vect{x} , \Vect{y} }\) is linearly independent.

  1. \(\Vect{x} = (1,0)\)

  2. \(\Vect{x} = (3,1)\)

  3. \(\Vect{x} = (2,2)\)

  4. \(\Vect{x} = (3,-5)\)

  5. \(\Vect{x} = (12,7)\)

  6. \(\Vect{x} = (x,y)\neq (0,0)\) is an arbitrary vector.

Problem 6

Given the vector \(\Vect{x}\) in \(\RNrSpc{3}\), find a vector \(\Vect{y}=(u,v,w)\) such that \(S\DefEq \Set{ \Vect{x} , \Vect{y} }\) is linearly independent.

  1. \(\Vect{x} = (1,0,2)\)

  2. \(\Vect{x} = (1,1,1)\)

  3. \(\Vect{x} = (2,-1,3)\)

  4. \(\Vect{x} = (3,2,-5)\)

  5. \(\Vect{x} = (5,-1,-4)\)

  6. \(\Vect{x}=(x,y,z)\neq (0,0,0)\) is arbitrary in \(\RNrSpc{3}\).

Problem 7

Show that, if the subset \(S\) of \(\RNrSpc{n}\) contains the \(\Vect{0}\)-vector, then \(S\) is linearly dependent.

Problem 8

If \(\Vect{x},\Vect{y}\) are nonparallel vectors in \(\RNrSpc{3}\), show that \(S\DefEq \Set{ \Vect{x},\Vect{y}, \CrssPr{ \Vect{x} }{ \Vect{y} } }\) is linearly independent.

Problem 9

Show that any collection of \((n+1)\) or more vectors in \(\RNrSpc{n}\) is linearly dependent.