2.1 and 2.2

Vector Addition and Scalar Multiplication

• To add two vectors, they must have the same number of entries (same dimension, i.e., live in the same $\mathbb{R}^n$!).

  • Coordinate-wise addition: if $\mathbf{w}=(w1,\dots,wn)$ and $\mathbf{v}=(v1,\dots,vn)$, then

    $\mathbf{w}+\mathbf{v}=(w1+v1,\dots, wn+vn)$ .

  • Example: $\mathbf{a}=(2,1,3), \quad \mathbf{b}=(-1,2,6) \Rightarrow \mathbf{a}+\mathbf{b}=(1,3,9)$ .

• Scalar multiplication by a real number $\alpha$ scales every entry of the vector:

  • If $\mathbf{v}=(v1,\dots,vn)$, then

    $\alpha\mathbf{v}=(\alpha v1,\dots, \alpha vn)$ .

• Example: $-10\,(2,-3,5)=(-20,30,-50)$ .

• Note: Later in the course we’ll discuss the dot product and the cross product; for now we focus on addition and scaling.

• Geometric intuition: vectors are free displacements; addition corresponds to chaining displacements (tip-to-tail or tail-to-tail via parallelogram).

Geometric Perspective: Addition, Scaling, and Subtraction

• Geometric rule for adding vectors: place the vectors with tails together, or equivalently use the parallelogram rule; the resulting vector goes from the origin to the far corner of the parallelogram.

• Scaling geometrically means stretching or shrinking the vector:

  • If (k>0), direction stays the same and length scales by $\Vert k\Vert$ .

  • If (k<0), the direction is flipped (mirror through the origin) and length scales by $\Vert k\Vert$ .

  • Examples: scale by $(1.5)$ (stretch), scale by $(0)$ (collapse to the origin), scale by $\pi$ (stretch by factor $\pi$); negative scale flips direction.

• Subtraction: $\mathbf{v}-\mathbf{w} = \mathbf{v}+(-\mathbf{w})$.

  • Geometric interpretation: the vector you add to $\mathbf{w}$ to land at $\mathbf{v}$ is $\mathbf{v}-\mathbf{w}$ . Equivalently, it’s the vector from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ .

• Note on drawing: vectors are often drawn from the origin for simplicity, but a vector is a displacement and does not depend on a fixed starting point.

Subtle Practice: Vector algebra vs geometry (example outline)

• Suppose you want to compute $\mathbf{V}1-\mathbf{V}2$ and illustrate geometrically.

  • Algebraic method: subtract components: if $\mathbf{V}1=(v{11},v{12})$ and $\mathbf{V}2=(v{21},v{22})$, then

    $\mathbf{V}1-\mathbf{V}2=(v{11}-v{21},\; v{12}-v{22})$.

  • Geometric method: compute $(-\mathbf{V}2)$ (the vector opposite to $\mathbf{V}2$) and add to $\mathbf{V}1$ (i.e., $\mathbf{V}1+(-\mathbf{V}_2)$); this gives the same result.

• Example (illustrative): take $\mathbf{V}1=(2,1)$ and $\mathbf{V}2=(-1,3)$ .

  • Algebraic: $\mathbf{V}1-\mathbf{V}2=(2-(-1),\;1-3)=(3,-2)$ .

  • Geometric: $(-\mathbf{V}2=(1,-3))$; then $\mathbf{V}1+(-\mathbf{V}_2)=(2,1)+(1,-3)=(3,-2)$ .

• Important geometric note: you can translate vectors as needed when describing the displacement; the vector itself is the same regardless of where you draw it.

• Quick encouragement: practice problems on vector addition, subtraction, and scaling to build geometric intuition.

Linear Combinations and the Span (Definition first real in course)

• Linear combination definition:

  • A vector $\mathbf{w}\in\mathbb{R}^n$ is a linear combination of vectors $\mathbf{v}1,\dots,\mathbf{v}p$ if there exist scalars $(c1,\dots, cp\in\mathbb{R})$ such that

    $\mathbf{w}=c1\mathbf{v}1+\cdots+cp\mathbf{v}p$ .

  • In simpler terms, a linear combination is $\mathbf{w} = c1\mathbf{v}1 + c2\mathbf{v}2 + \dots + cp\mathbf{v}p$ where each $c_i$ is a constant (scalar).

  • The idea: $\mathbf{w}$ can be reached by taking $(ci)$ steps in the direction of each $\mathbf{v}i$ (steps can be positive or negative).

• We’ll often consider linear combinations of two or three vectors to build intuition.

• Examples of other linear combinations: $\mathbf{v}1+\mathbf{v}2$, $(0\cdot\mathbf{v}1+1\cdot\mathbf{v}2)$, $\mathbf{v}1-\mathbf{v}2$, $(0\mathbf{v}1+0\mathbf{v}2)$ (the zero vector). All of these are linear combinations of $\mathbf{v}1,\mathbf{v}2$ .

The Span of a Set of Vectors

• Span notation and meaning:

  • For vectors $\mathbf{v}1,\dots,\mathbf{v}p$ in $\mathbb{R}^n$, the span is

    \mathrm{Span}{\mathbf{v}1,\dots,\mathbf{v}p}=\left{x1\mathbf{v}1+\cdots+xp\mathbf{v}p\;\middle|\;x_i\in\mathbb{R}\right} .

  • This is the set of all linear combinations of the given vectors.

• Examples:

  • If $\mathrm{Span}{\mathbf{v}1}$: all scalar multiples of $\mathbf{v}1$, i.e.,

    $\mathrm{Span}{\mathbf{v}1}={\lambda\mathbf{v}1\;|\;\lambda\in\mathbb{R}}$.

    Geometrically: a line through the origin in the direction of $\mathbf{v}_1$ .

  • The span always contains the zero vector (take all coefficients as zero).

• In the plane $\mathbb{R}^2$:

  • If $\mathbf{v}1$ and $\mathbf{v}2$ are not multiples of each other (not co-linear), then

    $\mathrm{Span}{\mathbf{v}1,\mathbf{v}2}=\mathbb{R}^2$ .

    Intuition: you can reach any point in the plane by appropriate linear combinations.

  • If $\mathbf{v}1$ and $\mathbf{v}2$ are parallel (collinear), the span is a line through the origin.

  • If both are the zero vector, the span is {0} (the origin only).

• In three dimensions (or higher):

  • The span of two non-parallel vectors is a plane through the origin (a 2D subspace).

  • The span of three vectors can be all of $\mathbb{R}^3$ if they are not all contained in a single plane through the origin (i.e., they are not coplanar or, equivalently, not all linear combinations lie in a 2D subspace).

  • More generally, the span of a set of vectors is the subspace consisting of all linear combinations; its dimension is at most the number of vectors and the ambient space dimension.

• Connection to the origin: every span is a subspace that passes through the origin; hence the origin is always included.

Span vs. Solving Linear Systems (algebra \Leftrightarrow geometry)

• Consider a system of equations that can be written in vector form as

  $x\mathbf{v}1 + y\mathbf{v}2 = \mathbf{b}$ ,

  where $\mathbf{v}1,\mathbf{v}2$ are fixed vectors and $(x,y)$ are unknowns, with $\mathbf{b}$ the right-hand side.

• This vector equation is consistent exactly when the right-hand side $\mathbf{b}$ lies in the span of $\mathbf{v}1$ and $\mathbf{v}2$:

  • i.e., when there exist scalars $(x,y)$ such that

    $\mathbf{b}=x\mathbf{v}1+y\mathbf{v}2$ .

  • In other words, the system has a solution if and only if $\mathbf{b}\in\mathrm{Span}{\mathbf{v}1,\mathbf{v}2}$ .

• Example outline (illustrative, not from a specific numeric problem):

  • Take $\mathbf{v}1=(1,2,6)$, $\mathbf{v}2=(-1,-2,-1)$, and $\mathbf{b}=(8,16,3)$ .

  • Row-reduce the augmented matrix (or solve the equation) to find scalars $(x,y)$ with

    $x\mathbf{v}1 + y\mathbf{v}2 = \mathbf{b}$ .

  • One solution is $(x=-1, y=-9)$, which shows that $\mathbf{b}$ lies in the span of $\mathbf{v}1,\mathbf{v}2$ .

  • Therefore, solving the vector equation is the same as solving the corresponding system of equations; the two perspectives are equivalent.

• Takeaways for exams:

  • Be able to state definitions exactly (linear combination, span).

  • Understand that solving a linear system can be viewed as asking whether a target vector lies in the span of certain vectors (columns of the coefficient matrix).

  • Expect word-for-word definitions or precise articulation of these concepts.

Quick Practical Connections and Takeaways

• The span is a subspace containing the origin, generated by all linear combinations of a given set of vectors.

• Geometrically:

  • Span of one nonzero vector: a line through the origin.

  • Span of two noncolinear vectors in $\mathbb{R}^n$: a plane through the origin.

  • Span of three vectors in $\mathbb{R}^3$: could be all of $\mathbb{R}^3$ if they are not coplanar; otherwise a plane or a line or {0} depending on dependence.

• Algebraically:

  • A vector is a linear combination of a set if it can be formed by a finite linear combination of those vectors with real coefficients.

  • The span is the set of all such linear combinations; it is the collection of all points you can reach by those displacements starting from the origin.

• In practice:

  • The vector equation $$(x1\mathbf{v}1+\