Vectors in R^n and the Standard Basis

Part 1, Chapter 1: Linear Algebra Toolkit

Learning objectives

Define $\mathbb{R}^n$ as the set of ordered n-tuples and perform component-wise addition and scalar multiplication
Form linear combinations and compute the span of a set of vectors
Distinguish linear independence from linear dependence and recognise the standard basis
Predict when a set of vectors must be dependent purely from a count of dimensions

Every linear-algebra problem you will ever encounter lives inside $\mathbb{R}^n$ , or a structure built to behave like it. Whether you are training a 768-dimensional language-model embedding, rotating a triangle in a game engine, or fitting a regression line to housing data, you are working inside an ordered tuple of real numbers, equipped with two operations: add two tuples component-wise, scale a tuple by a real number. Almost nothing else is required. The whole apparatus of vectors, span, independence, and basis, the entire subject of this chapter, is built on those two operations and nothing more.

What a vector actually is

The space $\mathbb{R}^n$ is the set of all ordered $n$ -tuples $\mathbf{v} = (x_1, x_2, \ldots, x_n)$ with each $x_i \in \mathbb{R}$ iinmathbbR. Two operations are built in: addition, defined as $\mathbf{u} + \mathbf{v} = (u_1 + v_1, \ldots, u_n + v_n)$ vn), and scalar multiplication, $c \mathbf{v} = (c x_1, \ldots, c x_n)$ . These obey eight axioms (associativity, commutativity, distributivity, etc.) that together make $\mathbb{R}^n$ a vector space. A vector has no preferred starting point, what matters is its components, which encode magnitude and direction simultaneously.

Linear combinations and span

A linear combination of vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ is any expression of the form $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k$ mathbfvk with $c_i \in \mathbb{R}$ iinmathbbR. The set of all such combinations is the span, written $\operatorname{span}(\mathbf{v}_1, \ldots, \mathbf{v}_k)$ . Geometrically, the span of one non-zero vector is a line through the origin; the span of two non-parallel vectors in $\mathbb{R}^3$ is a plane; in general, the span is the smallest subspace containing the given vectors.

Independence and the standard basis

Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent when the only solution to $c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k = \mathbf{0}$ mathbfvk=mathbf0 is $c_1 = \cdots = c_k = 0$ 0. If a non-trivial solution exists, the vectors are dependent, meaning at least one is a linear combination of the others, it carries no new information. The standard basis of $\mathbb{R}^n$ consists of the unit vectors $\mathbf{e}_1 = (1, 0, \ldots, 0), \mathbf{e}_2 = (0, 1, 0, \ldots, 0), \ldots, \mathbf{e}_n = (0, \ldots, 0, 1)$ (0,ldots,0,1), the cleanest possible independent spanning set.

The matrix-multiplier above shows how a $2 \times 2$ matrix maps the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2$ to new vectors that become the columns of the matrix. The image of the unit square is a parallelogram spanned by those two image vectors, an immediate geometric picture of span and independence in $\mathbb{R}^2$ . Try setting both columns equal: the parallelogram collapses to a line, telling you the columns are dependent and span only a 1-dimensional subspace.

Where this shows up

Machine learning embeddings: A modern transformer encodes each word or image patch as a vector in $\mathbb{R}^{768}$ or higher. Similar concepts cluster together, and operations like analogy ("king − man + woman") are literally vector addition in $\mathbb{R}^n$ .

Computer graphics: Every vertex of a 3D model is a point in $\mathbb{R}^3$ (or $\mathbb{R}^4$ with homogeneous coordinates). A scene with one million polygons stores millions of $\mathbb{R}^3$ vectors and operates on them all in parallel on the GPU.

Statistics and regression: A data set of $m$ observations of $n$ features is naturally a collection of vectors in $\mathbb{R}^n$ . Linear regression finds the best linear combination of the feature vectors that explains the response variable, pure span arithmetic.

Pause and think: If you have $n+1$ vectors sitting inside $\mathbb{R}^n$ , must at least one of them be a linear combination of the others? Use the geometric picture (a plane in $\mathbb{R}^2$ can hold at most two independent directions) before reaching for a formal proof.

Try it

Before computing, predict whether $(3, 5)$ lies in $\operatorname{span}\{(1, 2), (1, 1)\}$ . Then verify by solving $c_1 + c_2 = 3$ and $2 c_1 + c_2 = 5$ .

Predict: are $(1, 2)$ and $(2, 4)$ linearly independent? Then verify using the determinant of the $2 \times 2$ matrix whose columns are these vectors.

Express $(7, 11)$ as a linear combination of $(1, 3)$ and $(2, 1)$ . Predict the coefficients first by inspection, then solve the system precisely.

Use the matrix widget above: set the columns to $(2, 0)$ and $(1, 3)$ . Read off the area of the image parallelogram (it equals the absolute value of the determinant). Predict before you compute.

True or false: any three vectors in $\mathbb{R}^2$ are linearly dependent. Justify in one sentence using the dimension count.

A trap to watch for

Students routinely confuse "independent" with "non-zero" and "spans $\mathbb{R}^n$ " with "has $n$ vectors". Neither identification holds. The set $\{(1, 0), (2, 0)\}$ has two non-zero vectors but is dependent (the second is twice the first) and spans only the $x$ -axis. Conversely, $\{(1, 1), (1, -1), (1, 0)\}$ has three vectors in $\mathbb{R}^2$ and so must be dependent, you can never have more independent vectors than the dimension of the space. Always check both conditions explicitly: the vectors must span the space and be linearly independent for the count of vectors to equal the dimension.

What you now know

You can compute linear combinations, decide membership in a span, and test independence in $\mathbb{R}^n$ . The next section (§1.3) lifts these ideas one level: linear maps between $\mathbb{R}^n$ and $\mathbb{R}^m$ are completely encoded by matrices, and span / independence translate into the column space and kernel.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 1, §1.2.

Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press, ch. 1.

Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer, ch. 1-2.

Lay, D. C. (2015). Linear Algebra and Its Applications (5th ed.). Pearson, ch. 1.

Halmos, P. R. (1958). Finite-Dimensional Vector Spaces (2nd ed.). Van Nostrand, ch. 1.

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