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Vectors & Linear Combinations

Vectors & Linear Combinations

Chara Chara
October 1, 2025
2 min read
linear-algebra lecture-3
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Note (Extra Reading)

3Blue1Brown’s linear algebra series gives a visual intuition for Vectors. Start with this episode:

Review from Last Time

Pivots, Pivot Columns, Free Variables

  • Pivot: first nonzero entry in a nonzero row.
  • Pivot column: column with a pivot.
  • Free variable: non-pivot column (not the last column).
  • Number of pivots ≤\le number of rows.

Example

[$∗∗∗∗∗0$∗∗∗∗000$∗∗000000000000]\begin{bmatrix} \$ & * & * & * & * & * \\ 0 & \$ & * & * & * & * \\ 0 & 0 & 0 & \$ & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}

How to describe all solutions:

  1. Assign free variables arbitrary values.
  2. Express pivot variables in terms of free variables.

Example

[1100∣3/20010∣20001∣1/20000∣0]    ⇒    x1+x2=3/2x3=2x4=1/2\begin{bmatrix} 1 & 1 & 0 & 0 & | & 3/2 \\ 0 & 0 & 1 & 0 & | & 2 \\ 0 & 0 & 0 & 1 & | & 1/2 \\ 0 & 0 & 0 & 0 & | & 0 \end{bmatrix} \;\;\Rightarrow\;\; \begin{aligned} x_1 + x_2 &= 3/2 \\ x_3 &= 2 \\ x_4 &= 1/2 \end{aligned}

Let x2=sx_2 = s Then:

x1=32−s,    x2=s,    x3=2,    x4=12x_1 = \tfrac{3}{2} - s,\;\;x_2=s,\;\;x_3=2,\;\;x_4=\tfrac{1}{2}

Vectors

An mm-vector (column vector in Rm\mathbb{R}^m) is an m×1m\times 1 matrix:

a⃗=[a1a2a3⋮am]\vec{a}= \begin{bmatrix} a_1\\a_2\\a_3\\ \vdots \\a_m \end{bmatrix}
  • Addition:
[a1a2a3⋮am]+[b1b2b3⋮bm]=[a1+b1a2+b2a3+b3⋮am+bm]\begin{bmatrix} a_1\\a_2\\a_3\\\vdots\\a_m \end{bmatrix} + \begin{bmatrix} b_1\\b_2\\b_3\\\vdots\\b_m \end{bmatrix} = \begin{bmatrix} a_1+b_1\\a_2+b_2\\a_3+b_3\\\vdots\\a_m+b_m \end{bmatrix}
  • Scalar multiplication:
c[a1a2a3⋮am]=[ca1ca2ca3⋮cam]c\begin{bmatrix} a_1\\a_2\\a_3\\\vdots\\a_m \end{bmatrix} = \begin{bmatrix} ca_1\\ca_2\\ca_3\\\vdots\\ca_m \end{bmatrix}

Solutions as Vectors

Solutions of a linear system can be written as vectors.

Example

[x1x2x3x4x5]=[3/2−s−πts2−ett1/2]=[3/20201/2]+s[−11000]+t[−π0−e10]\begin{bmatrix} x_1\\x_2\\x_3\\x_4\\x_5 \end{bmatrix} = \begin{bmatrix} 3/2 - s - \pi t \\ s \\ 2 - e t \\ t \\ 1/2 \end{bmatrix} = \begin{bmatrix} 3/2 \\ 0 \\ 2 \\ 0 \\ 1/2 \end{bmatrix} + s\begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + t\begin{bmatrix} -\pi \\ 0 \\ -e \\ 1 \\ 0 \end{bmatrix}

And the left hand side expression is called a linear combination.

Linear Combination

Definition (Linear Combination)

Given vectors v⃗1,…,v⃗k\vec{v}_1,\dots,\vec{v}_k and scalars s1,…,sks_1,\dots,s_k:

y⃗=s1v⃗1+⋯+skv⃗k\vec{y}=s_1\vec{v}_1+\cdots+s_k\vec{v}_k

is called a linear combination of {v⃗1,…,v⃗k}\{\vec{v}_1,\dots,\vec{v}_k\}

Examples

  1. Given a vector v⃗\vec{v}, a linear combination of v⃗\vec{v} is just a scalar multiple -> av⃗a\vec{v}

  2. Given vectors:

v⃗1=[12],    v⃗2=[34],    y⃗=[56]\vec{v}_1=\begin{bmatrix}1\\2\end{bmatrix},\;\; \vec{v}_2=\begin{bmatrix}3\\4\end{bmatrix},\;\; \vec{y}=\begin{bmatrix}5\\6\end{bmatrix}

Check if y⃗\vec{y} is a linear combination of v⃗1,v⃗2\vec{v}_1,\vec{v}_2:

Find s1,s2s_1,s_2 such that

[56]=s1[12]+s2[34]\begin{bmatrix}5\\6\end{bmatrix} = s_1\begin{bmatrix}1\\2\end{bmatrix} + s_2\begin{bmatrix}3\\4\end{bmatrix}

Equivalent to solving:

[12]x1+[34]x2=[56]\begin{bmatrix}1\\2\end{bmatrix}x_1+ \begin{bmatrix}3\\4\end{bmatrix}x_2= \begin{bmatrix}5\\6\end{bmatrix}