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Vectors and Linear Combinations

What is a Linear Combination?

A linear combination of two vectors means multiplying them by numbers (called scalars) and adding the results. For example:

\[ 3v+5w \]

This is a typical linear combination of the vectors v and w.

If

\[ \mathbf{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]

then

\[ 3\mathbf{v} + 5\mathbf{w} = 3\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 5\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \end{bmatrix} + \begin{bmatrix} 0 \\ 5 \end{bmatrix} = \begin{bmatrix} 3 \\ 11 \end{bmatrix} \]

This means we move 3 units across (in x direction) and 11 units up (in y direction).

Visualizing Vectors

A vector like \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\)goes 2 steps in x and 3 steps in y. It points to the location (2, 3) in the xy-plane.

All combinations like \(\mathbf{v} + d\mathbf{w}\) (fill in every point in the plane) if v and w point in different directions. They cover the whole 2D space.

In 3D space, a combination like

\[ c\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + d\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} \]

fills a plane in xyz space (but not the full space unless we add a third independent vector).

Vector Operations

A vector has multiple components, like:

\[ \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \]

We treat v as a single object, not two separate numbers.

Vector Addition

Add corresponding parts:

\[ \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \]

Then:

\[ \mathbf{v} + \mathbf{w} = \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix} \]

Vector Subtraction

Same idea:

\[ \mathbf{v} - \mathbf{w} = \begin{bmatrix} v_1 - w_1 \\ v_2 - w_2 \end{bmatrix} \]

Scalar Multiplication

Multiply every part of a vector by a number:

\[ \mathbf{v} = 2\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 2v_1 \\ 2v_2 \end{bmatrix}, \quad -\mathbf{v} = \begin{bmatrix} -v_1 \\ -v_2 \end{bmatrix} \]

A scalar is just a number like 2 or -1.

The sum of a vector and its negative gives the zero vector:

\[ \mathbf{v} + (-\mathbf{v}) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]

Linear Combinations in Practice

A linear combination means: \(c\mathbf{v} + d\mathbf{w}\)

You can also get:

  • Sum: \(1v+1w\)
  • Difference: \(1vāˆ’1w\)
  • Zero vector: \(0v+0w\)
  • Vector \(cv\) in the direction of \(v\): \(cv+0wc\)

All of these are valid linear combinations.

Representing Vectors

There are three main ways to describe a vector v:

  • By its components: (4, 2)
  • As an arrow from (0, 0) to (4, 2)
  • As a point at (4, 2) in the plane

You can visualize v + w as going along v, then along w, or directly along the diagonal from start to finish.

Vectors in 3D

A 3D vector looks like: \(\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}\)This points from \((0, 0, 0)\) to \((vā‚, vā‚‚, vā‚ƒ)\) in 3D space.

Writing \(\mathbf{v} = (1, 1, -1)\) is a shortcut for the column vector\(\begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}\)It is not a row vector which is the transpose.

Addition in 3D works the same way: \(\mathbf{v} + \mathbf{w} = \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \\ v_3 + w_3 \end{bmatrix}\)