Linear Combinations

All of linear algebra is built on the two basic operations on vectors: addition and scalar multiplication.

When we combine these two operations, we form a linear combination of vectors. For now, let’s stick to an example with only two vectors.

The sum of cv⃗c\vec{v} and dw⃗d\vec{w} - where cc and dd are scalars - is a linear combination of v⃗\vec{v} and w⃗\vec{w}

Special Linear Combinations

If we choose some specific values for cc and dd, we can reduce a linear combination to concepts already covered in the course:

  • 1v⃗+1w⃗1\vec{v} + 1\vec{w} is just the sum of v⃗\vec{v} and w⃗\vec{w},
  • 1v⃗−1w⃗1\vec{v} - 1\vec{w} is their difference,
  • 0v⃗+0w⃗0\vec{v} + 0\vec{w} is the zero vector,
  • cv⃗+0w⃗c\vec{v} + 0\vec{w} is just the vector v⃗\vec{v} multiplied by the scalar cc.

Example

Let me illustrate by forming an arbitrary linear combination of two arbitrary vectors:

v⃗=[23],w⃗=[14],c=2,d=3\vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \vec{w} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}, c = 2, d = 3 cv⃗+dw⃗=[c⋅v1+d⋅w1c⋅v2+d⋅w2]=[2⋅2+3⋅12⋅3+3⋅4]=[718]\begin{align*} c\vec{v} + d\vec{w} &= \begin{bmatrix} c \cdot v_1 + d \cdot w_1 \\ c \cdot v_2 + d \cdot w_2 \end{bmatrix} \\ &= \begin{bmatrix} 2 \cdot 2 + 3 \cdot 1 \\ 2 \cdot 3 + 3 \cdot 4 \end{bmatrix} \\ &= \begin{bmatrix} 7 \\ 18 \end{bmatrix} \end{align*}

So the vector [718]\begin{bmatrix} 7 \\ 18 \end{bmatrix} is one of the possible linear combinations of v⃗\vec{v} and w⃗\vec{w}.