Key Definitions & Concepts — EM-II Unit I
Quick-reference glossary for all important terms encountered in Unit I (Linear Algebra). Click any heading to jump to its detailed explanation.
Ordered Basis
The everyday analogy
Think of a locker row with numbered slots: Slot 1, Slot 2, Slot 3, …
- A basis is like saying “these 3 lockers exist” — you know which lockers, but nobody assigned numbers to them.
- An ordered basis is like saying “Locker A is Slot 1, Locker B is Slot 2, Locker C is Slot 3.” Now every item can be described by its slot number.
A coordinate vector is just the list of slot numbers. If you renumber the slots (change the order), the same items get different slot numbers — even though nothing physically moved.
Formal definition
An ordered basis of a vector space $V$ is a basis whose vectors are listed in a fixed sequence. Two ordered bases containing the same vectors in a different order are considered different.
\[B = (\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n) \quad \neq \quad B' = (\vec{v}_2, \vec{v}_1, \ldots, \vec{v}_n) \quad \text{(even though same set)}\]Notice the round brackets $(\;)$ — they denote an ordered tuple, unlike the curly braces $\lbrace\;\rbrace$ used for unordered sets.
Why does the order matter?
Step 1 — A basis alone just says “you can build any vector.” If $\lbrace \vec{v}_1, \vec{v}_2 \rbrace$ is a basis of $V$, then every $\vec{x} \in V$ can be written uniquely as:
\[\vec{x} = c_1\vec{v}_1 + c_2\vec{v}_2\]Step 2 — But to store those coefficients, you need to decide which coefficient goes first. We package them into a column:
\[[\vec{x}]_B = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}\]This only makes sense if we agreed that “first entry = coefficient of $\vec{v}_1$” and “second entry = coefficient of $\vec{v}_2$.” That agreement is the ordering.
Step 3 — Swap the order, and the same vector gets different coordinates:
\[B = (\vec{v}_1, \vec{v}_2) \;\Rightarrow\; [\vec{x}]_B = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}, \qquad B' = (\vec{v}_2, \vec{v}_1) \;\Rightarrow\; [\vec{x}]_{B'} = \begin{pmatrix} c_2 \\ c_1 \end{pmatrix}\]The vector $\vec{x}$ hasn’t changed. The building blocks haven’t changed. Only the labeling (“who is #1, who is #2”) changed — and that’s enough to shuffle the entire coordinate vector.
Concrete example
Let $\vec{x} = (5, 3)$ in $\mathbb{R}^2$.
Ordered basis 1: $B = ((1,0),\; (0,1))$ — the standard basis in the usual order.
\[\vec{x} = \underbrace{5}_{c_1}(1,0) + \underbrace{3}_{c_2}(0,1) \quad\Longrightarrow\quad [\vec{x}]_B = \begin{pmatrix} 5 \\ 3 \end{pmatrix}\]Ordered basis 2: $B’ = ((0,1),\; (1,0))$ — same vectors, swapped order.
\[\vec{x} = \underbrace{3}_{c_1'}(0,1) + \underbrace{5}_{c_2'}(1,0) \quad\Longrightarrow\quad [\vec{x}]_{B'} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}\]| Ordered Basis | 1st basis vector | 2nd basis vector | Coordinate Vector |
|---|---|---|---|
| $B = ((1,0),\; (0,1))$ | $(1,0)$ | $(0,1)$ | $(5,\; 3)^T$ |
| $B’ = ((0,1),\; (1,0))$ | $(0,1)$ | $(1,0)$ | $(3,\; 5)^T$ |
Same set of basis vectors, different order → different coordinates → different transition matrices, different matrix representations of maps — everything downstream changes.
Where ordered bases are essential
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Coordinate vectors — the $i$-th entry of $[\vec{x}]_B$ is the coefficient of the $i$-th vector in the ordered list. Without ordering, you can’t assign entries to positions.
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Transition matrices — $P_{B \to B’}$ converts coordinates from $B$ to $B’$. Each column $j$ of $P$ is the coordinate vector of the $j$-th old basis vector in the new basis. Without a fixed “which is $j$-th?”, the matrix can’t be built.
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Matrix representations of linear maps — writing $[T]_B$ as a matrix requires knowing which column corresponds to which input basis vector and which row to which output basis vector. Order defines both.
Bottom line: “Basis” tells you what the building blocks are. “Ordered basis” tells you what label each building block gets. Every matrix-based construction in linear algebra (coordinates, change-of-basis, map representations) needs those labels.