Show that any collection of two or more vectors in \(\RNr\) is linearly dependent.
To see that any collection of two or more vectors in \(\RNr\) is linearly dependent we use that part iii. of the lemma. Then we observe that \(\RNr\) is spanned by a single vector, for example \(\StdBssVec{1} = (1)\). So if any collection of vectors in \(\RNr\) is to be linearly independent, the number of vectors in the collection must be 0 or 1. Accordingly, a collection of two or more vectors in \(\RNr\) is linearly dependent.
Show that any collection of three or more vectors in \(\RNrSpc{2}\) is linearly dependent.
To see that any collection of three or more vectors in \(\RNrSpc{2}\) is linearly dependent we use that part iii. of the lemma. Indeed, we observe that \(\RNrSpc{2}\) is spanned by a two vectors, for example \(\StdBssVec{1} = (1,0)\) and \(\StdBssVec{2}=(0,1)\). This is so because every vector \(\Vect{x}=(a,b)\) in \(\RNrSpc{2}\) may be expressed as the linear combination
| \((a,b)\) | \(=\) | \(a\cdot \StdBssVec{1} + b\cdot \StdBssVec{2}\) |
So if any collection of vectors in \(\RNrSpc{2}\) is to be linearly independent, the number of vectors in the collection must be less than or equal to 2. Accordingly, a collection of three or more vectors in \(\RNrSpc{2}\) is linearly dependent.
Show that any collection of four or more vectors in \(\RNrSpc{3}\) is linearly dependent.
To see that any collection of four or more vectors in \(\RNrSpc{3}\) is linearly dependent we use that part iii. of the lemma. Indeed, we observe that \(\RNrSpc{3}\) is spanned by a three vectors, for example \(\StdBssVec{1} = (1,0,0)\), \(\StdBssVec{2}=(0,1,0)\) and \(\StdBssVec{3}=(0,0,1)\). This is so because every vector \(\Vect{x}=(a,b,c)\) in \(\RNrSpc{3}\) may be expressed as the linear combination
| \((a,b,c)\) | \(=\) | \(a\cdot \StdBssVec{1} + b\cdot \StdBssVec{2} + c\cdot \StdBssVec{3}\) |
So if any collection of vectors in \(\RNrSpc{3}\) is to be linearly independent, the number of vectors in the collection must be less than or equal to 3. Accordingly, a collection of four or more vectors in \(\RNrSpc{3}\) is linearly dependent.