How do you multiply column vectors by a scalar?

To multiply a column vector by a scalar, you take each element of the vector and multiply it by that scalar. This means if you have a column vector with two elements, each of those elements is independently multiplied by the scalar.

For example, if you have a column vector represented as:

[ \begin{pmatrix} a \ b \end{pmatrix} ]

and you want to multiply it by a scalar ( k ), the operation is performed as follows:

[ k \cdot \begin{pmatrix} a \ b \end{pmatrix} = \begin{pmatrix} k \cdot a \ k \cdot b \end{pmatrix} ]

This means you multiply the top element ( a ) by the scalar ( k ) and the bottom element ( b ) by the scalar ( k ) as well. This confirms that the process applies to both elements of the column vector, leading to a scaled vector.

This method reflects the principle of scalar multiplication in linear algebra, where each component of the vector is affected by the scalar, thereby producing a new vector with each original element scaled accordingly.