Linear Combination

Definition: Vectors

Let

• $\mathbf{w}$ and ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_4$ : Vectors in $R^n$
• $k_1,k_2,\dots ,k_r$ : Scalars

If

$$\mathbf{w}=k_1{\mathbf{v}}_1+k_2{\mathbf{v}}_2+\dots ,k_r{\mathbf{v}}_r$$

Then

• We say $\mathbf{w}$ is a linear combination of ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_r$
• We call $k_1,k_2,\dots ,k_r$ the coefficients of the linear combination

Definition: Matrices

Let

• $B$ and $A_1,A_2,\dots ,A_r$ : Matrices of the same size
• $c_1,c_2,\dots ,c_r$ : Scalars

If

$$B=c_1{A}_1+c_2{A}_2+\cdots +c_r{A}_r$$

Then

• We say $B$ is a linear combination of $A_1,A_2,\dots A_r$
• We call $c_1,c_2,\dots ,c_r$ the coefficients of the linear combination