Every problem this course studies has some number of decision variables -- one, two, or several thousand -- and writing "x_1,x_2,\dots,x_n" out in full every single time would bury every later idea under bookkeeping. This section fixes one notational convention and uses it for the rest of the course: a full choice of every decision variable, collected into a single vector, is a design point, and several design points stacked together form a matrix. Nothing here is a new mathematical fact -- design points and the matrices built from them use ordinary vector and matrix operations, addition, scalar multiplication, stacking rows into a matrix. What's new is just the name this course gives one specific way of using them.
A single decision variable is one number you get to choose -- how many units to build, how large to set one parameter. Almost no real problem has only one of those, so this course needs a single word for "the whole bundle of choices, made all at once." That's what a design point is: not a new kind of object, just an ordinary vector, with the name chosen to remind you it represents one complete, specific decision rather than a direction or a displacement. Picture a recipe with several ingredient amounts as one design point \mathbf x=(x_1,x_2,x_3) -- flour, sugar, butter, all three chosen together, never one without the other two. If you had several candidate recipes to compare side by side, stacking them as rows of a table (flour, sugar, butter, one row per recipe) is precisely what stacking design points into a matrix means, and it lets every recipe's three numbers travel together as a single row instead of three separate lists.
A design point \mathbf x=(x_1,x_2,\dots,x_n) collects a complete choice of every decision variable into one vector in \mathbb R^n -- "design" because this is the language optimization borrows from engineering design, where each x_i might be a physical dimension, a resource quantity, or a model parameter being chosen. Ordinary vector notation applies without exception: components add and scale componentwise, \mathbf x+\mathbf y=(x_1+y_1,\dots,x_n+y_n), and c\mathbf x=(cx_1,\dots,cx_n). When a problem needs to track several design points at once -- several candidate solutions, or several data points a model is being fit to -- this course stacks them as the rows of a matrix X, an m\times n matrix for m design points in n variables each. This is exactly Linear Algebra's own design matrix convention, reused here under the same name rather than a fresh one, since it already means the right thing: one row per observation, one column per variable.
Two design points, each a bundle of choices
Each arrow is one design point -- a vector collecting every decision variable's value into a single object. Drag either one and watch its own components update together, never independently of each other.
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A design point with three decision variables
u is a single design point x=(2,1,3), three decision-variable values collected into one vector -- drag it and watch all three components move together, since a design point is one choice, not three separate ones.
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Three design points, stacked into a matrix
B's three rows are three separate design points in \mathbb R^2; A is B transposed. Step through the multiplication and watch each entry of the result form as a dot product between two design points at once.
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Worked example
Evaluating an objective at a design point
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1
For the design point \mathbf x=(3,-1,2) and the matrix A=\begin{pmatrix}1&0&2\\-1&3&1\end{pmatrix}, compute A\mathbf x.
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A\mathbf x is a matrix-vector product: row 1 of A dotted with \mathbf x, then row 2 dotted with \mathbf x.
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3this is ordinary matrix-vector multiplication -- design point or not, nothing about the computation changes
Row 1: (1)(3)+(0)(-1)+(2)(2)=3+0+4=7. Row 2: (-1)(3)+(3)(-1)+(1)(2)=-3-3+2=-4.
this is ordinary matrix-vector multiplication -- design point or not, nothing about the computation changes
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4
\boxed{A\mathbf x=(7,-4)} One design point, three decision-variable values, transformed into a two-component output by a single matrix multiplication -- exactly how this course will describe a linear constraint or a linear model applied to an entire design point at once.
Where this goes next
Word problems and modeling, from chapter 1 onward
Every optimization word problem in this course starts the same way: name the decision variables, then immediately collect them into one design point \mathbf x so the objective and every constraint can be written as a single function of \mathbf x rather than a list of separate variables. A production-planning problem with twelve products doesn't get twelve separate objective functions -- it gets one function f(\mathbf x) of a twelve-component design point, and one matrix A encoding every resource constraint at once. Fixing that convention here means chapter 1 can lean on it from its very first modeling example.
Check your understanding (counts toward your progress)
Several design points need to be tracked at once. What does this course do with them?
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Adding every design point together would destroy the ability to tell them apart -- stacking as rows keeps each one intact and separately readable.
Discarding every design point but the first and last throws away real information a later computation might need.
Collapsing a whole set of vectors into one number would lose almost everything about each individual design point.
Why
Stacking design points as the rows of a matrix X preserves every one of them individually while still letting the whole collection be manipulated as a single object.
For design points \mathbf x=(4,1) and \mathbf y=(-2,3), what is the second component of \mathbf x+\mathbf y?
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Why
(\mathbf x+\mathbf y)_2=1+3=4.
Correct answer
Also accepted:
Why
Why does this course insist on collecting every decision variable into a single design point \mathbf x rather than writing f(x_1,x_2,\dots,x_n) with every variable spelled out?
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Spelling out every variable is perfectly correct, just unwieldy -- the design-point convention is a notational convenience, not a correctness fix.
A design point can have any number of components -- two, twenty, or two thousand -- the notation doesn't restrict the count at all.
Collecting variables into a vector says nothing about whether the resulting function is linear or nonlinear -- that depends entirely on the function itself.
Why
The design-point convention buys scalability: f(\mathbf x) reads and is written the same way whether the underlying problem has two decision variables or two thousand.
In a design matrix X with m rows and n columns, what does m represent?
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That's what n, the column count, represents -- one column per decision variable, not one row.
The total entry count is m\\times n, not m alone -- m specifically counts the rows.
A design matrix's row count tracks design points, not constraints -- those are two entirely separate structures in this course.
Why
By this course's convention, each of a design matrix's m rows is one full design point, and each of its n columns is one decision variable.
What is a design point in this course's vocabulary?
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A single value is just one decision variable -- a design point is the FULL bundle of every decision variable's value, collected together.
A design point can be any candidate solution, optimal or not -- calling only the winner a design point would leave every other candidate without a name.
A design point can sit anywhere in (or even outside) the feasible region -- it isn't restricted to the boundary.
Why
A design point is any complete vector \mathbf x=(x_1,\dots,x_n) of decision-variable values, whether or not it happens to be feasible or optimal.
Check your understanding (counts toward your progress)
For design point \mathbf x=(-4,6,2), what is the second component of -0.5\mathbf x?
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Why
(-0.5\mathbf x)_2=-0.5(6)=-3.
Correct answer
Also accepted:
Why
For \mathbf x=(5,0,-3) and \mathbf y=(-2,4,1), what is the third component of \mathbf x+\mathbf y?
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Why
Adding third components directly: -3+1=-2.
Correct answer
Also accepted:
Why
For \mathbf x=(3,-2,1), what is the first component of 2\mathbf x?
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Why
(2\mathbf x)_1=2(3)=6.
Correct answer
Also accepted:
Why
For A=\begin{pmatrix}2&-1\\0&3\end{pmatrix} and design point \mathbf x=(4,2), compute the sum of both components of A\mathbf x.
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Why
A\mathbf x=((2)(4)+(-1)(2),\ (0)(4)+(3)(2))=(8-2,\ 0+6)=(6,6); the sum of both components is 6+6=12.
Correct answer
Also accepted:
Why
Using A=\begin{pmatrix}1&2\\3&0\end{pmatrix} and \mathbf x=(2,-1) from the previous question, what is the second component of A\mathbf x?
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Why
The second output component comes from A's second row: (3)(2)+(0)(-1)=6+0=6.
Correct answer
Also accepted:
Why
Practice: Design points and matrices, computed directly (3 problems · doesn’t affect your progress)
For design points \mathbf x=(6,-2) and \mathbf y=(1,5), what is the first component of \mathbf x-\mathbf y?
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Why
(\mathbf x-\mathbf y)_1=6-1=5.
Correct answer
Also accepted:
Why
For A=\begin{pmatrix}2&1\\1&3\end{pmatrix} and design point \mathbf x=(1,2), what is the second component of A\mathbf x?
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Why
Row 2 of A dotted with \mathbf x: (1)(1)+(3)(2)=1+6=7.
Correct answer
Also accepted:
Why
A design matrix has 4 design points, each with 6 decision variables. What are its dimensions?
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That would put variables in rows and design points in columns -- the opposite of this course's row-per-design-point convention.
Flattening into a single column would throw away the whole point of keeping each design point's components together.
Nothing here suggests the row and column counts happen to match -- 4 design points and 6 variables are two different numbers.
Why
One row per design point (4 rows) and one column per decision variable (6 columns) gives a 4\times6 matrix.
4\\times6