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Section 11.5 Theory
Subsection 11.5.1 Vector algebra
Here we list the basic rules of algebra for vectors in
\(\R^n\text{.}\) There is no need to prove these rules like we did for the rules of matrix algebra in
Subsection 4.5.1 , because we know from
Discovery 11.12 that vectors in
\(\R^n\) can be converted into column matrices and then the vector operations of addition, negation, and scalar multiplication all work as with matrices. And so, since the following rules are all valid when the vectors are replaced by column matrices, they are all valid for vectors in
\(\R^n\text{.}\)
Proposition 11.5.1 . Rules of vector algebra in \(\R^n\) .
The following are valid rules of vector algebra. In each statement, assume that \(\uvec{u},\uvec{v},\uvec{w}\) are arbitrary vectors and \(\zerovec\) is a zero vector, all of the same dimension. Also assume that \(k\) and \(m\) are scalars.
Rules of vector addition.
\(\displaystyle \uvec{v}+\uvec{u} = \uvec{u}+\uvec{v}\)
\(\displaystyle \uvec{u} + (\uvec{v} + \uvec{w}) = (\uvec{u} + \uvec{v}) + \uvec{w}\)
Rules involving scalar multiplication.
\(\displaystyle k(\uvec{u}+\uvec{v}) = k\uvec{u} + k\uvec{v}\)
\(\displaystyle (k+m)\uvec{v} = k\uvec{v} + m\uvec{v}\)
\(\displaystyle k(m\uvec{v}) = (km)\uvec{v}\)
\(\displaystyle 1\uvec{v} = \uvec{v}\)
\(\displaystyle (-1)\uvec{v} = -\uvec{v}\)
\(\displaystyle \uvec{u} - \uvec{v} = \uvec{u} + (-1)\uvec{v}\)
Rules involving a zero vector.
\(\displaystyle \uvec{v} + \zerovec = \uvec{v}\)
\(\displaystyle \uvec{v}-\uvec{v} = \zerovec\)
\(\displaystyle k\zerovec = \zerovec\)
