Linear combination

Given vectors u₁,u₂,...,u_m in Rⁿ and scalars k₁,k₂,...,k_m in R, the linear combination is a vector $$ v = k_1 u_1 + k_2 u_3 + ... + k_m v_m $$

Linear combination is the sum of the scalar products.

Example

Given 2 vectors in R³

$$ u_1 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} $$

$$ u_2 = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} $$

and 2 scalars in R

$$ k_1 = 3 $$

$$ k_2 = -1 $$

The linear combination is a vector v such as

$$ v = k_1 u_1 + k_2 u_2 $$

$$ v = 3 \cdot u_1 + (-1) \cdot u_2 $$

$$ v = 3 \cdot \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + ( -1 ) \cdot \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} $$

$$ v = \begin{pmatrix} 3 \cdot 1 \\ 3 \cdot 2 \\ 3 \cdot 1 \end{pmatrix} + \begin{pmatrix} ( -1 ) \cdot 3 \\ ( -1 ) \cdot (-1) \\ ( -1 ) \cdot 2 \end{pmatrix} $$

$$ v = \begin{pmatrix} 3 \\ 6 \\ 3 \end{pmatrix} + \begin{pmatrix} -3 \\ 1 \\ -2 \end{pmatrix} $$

$$ v = \begin{pmatrix} 3 + (-3) \\ 6 + 1 \\ 3 + (-2) \end{pmatrix} $$

$$ v = \begin{pmatrix} 0 \\ 7 \\ 1 \end{pmatrix} $$