Vectors and linear combinations

Lecture 2

Minjae Park

Auburn University
MATH 2660 - Spring 2026

January 9, 2026

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Vectors in \(\mathbb{R}^n\)

\[ \vec{v}=\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix} \]

Euclidean space

For some \(n\in\mathbb{N}\), the Euclidean space of dimension \(n\) is the set of all \(n\)-tuples \[ (a_1,a_2,\dots,a_n) \] where each \(a_i\in\mathbb{R}\).

Cartesian coordinate system

  • By interpreting each entry as a coordinate, we may interpret \((a_1,a_2,\dots,a_n)\) as a point in \(n\)-dimensional space.
  • For example, when \(n=2\), \((3,-1)\) is a point on the plane, and when \(n=3\), \((-1,0,2)\) is a point in 3-dimensional space.

Plot \((2,-3)\) on the plane.

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How we locate a point

Unconsciously, when we are given coordinates, we do something like this:

  • Locate the origin \((0,0)\).
  • Move \(2\) units in the \(x\) direction.
  • Move \(-3\) units (i.e., \(3\) units downward) in the \(y\) direction.

The last two steps are interchangeable: the order does not matter.

What if…

  • We change the origin?
  • We swap the \(x\) and \(y\) axes?
  • We tilt the axes?
  • We change the unit (inches vs. centimeters)?
  • These questions lead to the idea of vectors and linear combinations.

Coordinates for movement

We now interpret \((2,-3)\) as an instruction for movement:

  • Let the “unit step in the \(x\) direction” be \(\vec{e}_1\) and the “unit step in the \(y\) direction” be \(\vec{e}_2\). Then \[ (2,-3)=2\vec{e}_1-3\vec{e}_2. \]
  • In words: “move \(2\) units in the \(\vec{e}_1\) direction and \(-3\) units in the \(\vec{e}_2\) direction.”

  • A unit step in each direction can be naturally represented by \(\vec{e}_1=(1,0)\) and \(\vec{e}_2=(0,1)\).
  • With this interpretation, the previous arithmetic agrees with ordinary coordinate-by-coordinate algebra!
  • In particular, it makes sense to write \[ (2,-3)=2*(1,0)-3*(0,1). \] where the operations are carried out coordinate by coordinate.

Points vs. vectors

  • Consider two coordinate pairs: \((2,-3)\) and \((1,1)\).
  • Does it make sense to add two points?
  • It makes sense if we interpret \((a_1,a_2)\) as an instruction for movement (a vector).
  • As vectors, \((2,-3)+(1,1)\) means: do the movement \((2,-3)\), then do the movement \((1,1)\): \((2,-3)+(1,1)=(3,-2)\).

Column vector representation

To emphasize that this object represents a vector, rather than a point, we often write \[ \vec{v}=\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix}\qquad\text{or}\qquad \vec{v} = \langle a_1, a_2, \dots, a_n\rangle. \] We write vectors vertically, as columns, for convenience and consistency—this choice will become clear later in the course.

Arrow representations of a vector

  • If we start at the origin and follow \(\vec{v}\), the endpoint is the point with coordinates \((a_1,\dots,a_n)\).
  • So we often draw \(\vec{v}\) as an arrow from the origin to that point.
  • But a vector is an instruction for movement, so we can start from any point, not just the origin.
  • Therefore, many different arrows (same direction and length) represent the same vector.

Scalars and vectors

In examples like \((2,-3)=2\vec{e}_1-3\vec{e}_2\), there are two kinds of quantities: the magnitudes (\(2\) and \(-3\)) and the directions (\(\vec{e}_1\) and \(\vec{e}_2\)).

  • A scalar is a real number \(c\in\mathbb{R}\) (a magnitude).
  • A vector \(\vec{v}\) represents both a direction and a magnitude.

Elementary linear combination

In general, the vector \(\langle a_1,a_2,\dots,a_n \rangle\) in \(\mathbb{R}^n\) is the movement along each axis direction by the given magnitudes.

Equivalently, \[ \langle a_1,a_2,\dots,a_n \rangle =a_1\vec{e}_1+a_2\vec{e}_2+\cdots+a_n\vec{e}_n, \] where \(\vec{e}_i\) is the unit vector in the \(i\)th coordinate direction (the standard basis vector).

Vector operations

We define two fundamental operations for vectors \(\vec{v},\vec{w}\in\mathbb{R}^n\) and a scalar \(c\in\mathbb{R}\).

1. Vector addition \[ \vec{v}+\vec{w} = \begin{bmatrix}v_1\\ \vdots\\ v_n\end{bmatrix} + \begin{bmatrix}w_1\\ \vdots\\ w_n\end{bmatrix} = \begin{bmatrix}v_1+w_1\\ \vdots\\ v_n+w_n\end{bmatrix}. \]

Vector operations (continued)

2. Scalar multiplication \[ c\vec{v} = c\begin{bmatrix}v_1\\ \vdots\\ v_n\end{bmatrix} = \begin{bmatrix}cv_1\\ \vdots\\ cv_n\end{bmatrix}. \]

Geometric viewpoint of vector operations

  • 1. Vector addition: placing two vectors head-to-tail; equivalently, the sum is given by the diagonal of the parallelogram formed by the two vectors.

  • 2. Scalar multiplication:

    • If \(|c| > 1\), the vector is stretched.
    • If \(|c| < 1\), the vector is shrunk.
    • If \(|c| = 1\), the vector’s magnitude remains unchanged.

    If \(c < 0\), the direction of the vector is reversed.

Linear combinations

Suppose we are given vectors \(\vec{v}_1,\dots,\vec{v}_k\). A vector \(\vec{w}\) of the form \[ \vec{w}=c_1\vec{v}_1+\cdots+c_k\vec{v}_k \] is called a linear combination of \(\vec{v}_1,\dots,\vec{v}_k\) with coefficients \(c_1,\dots,c_k\in\mathbb{R}\).

Example

Let \[ \vec{v}_1=\begin{bmatrix}2\\-3\end{bmatrix}, \qquad \vec{v}_2=\begin{bmatrix}1\\0.5\end{bmatrix}. \] Compute \(\vec{w}=3\vec{v}_1-2\vec{v}_2\).

First, \[ 3\vec{v}_1 = 3\begin{bmatrix}2\\-3\end{bmatrix} = \begin{bmatrix}6\\-9\end{bmatrix}. \]

Example (continued)

Next, \[ -2\vec{v}_2 = -2\begin{bmatrix}1\\0.5\end{bmatrix} = \begin{bmatrix}-2\\-1\end{bmatrix}. \]

So \[ \vec{w} = \begin{bmatrix}6\\-9\end{bmatrix} + \begin{bmatrix}-2\\-1\end{bmatrix} = \begin{bmatrix}4\\-10\end{bmatrix}. \]