Span, Independence and Bases

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What a set of vectors can generate

The span of vectors is the set of all their linear combinations. A list is linearly independent when the only combination producing zero uses all zero coefficients. Dependence means at least one listed vector is redundant: it can be written using the others.

A basis of a vector space is an independent list that spans the space. Every vector in that space has exactly one coordinate list relative to the basis. Dimension is the number of vectors in any basis, not the number of vectors in an arbitrary generating list.

Visual guide

VISUAL GUIDEIndependent directions fill the plane

Basis direction (1, 1)

Vectors (1, 0) and (1, 1) form a basis: every point can be reached with a unique pair of coefficients. The point (3, 2) equals 1·(1, 0) + 2·(1, 1). Two parallel vectors would only span a line.

Worked example: test independence

The vectors $\mathbf v_1=(1,0,1)^T$ , $\mathbf v_2=(0,1,1)^T$ , and $\mathbf v_3=(1,1,2)^T$ are dependent because $\mathbf v_3=\mathbf v_1+\mathbf v_2$ . The first two are independent: the first two components of $c_1\mathbf v_1+c_2\mathbf v_2=0$ force $c_1=c_2=0$ .

They form a basis of the plane $z=x+y$ . The plane has dimension two even though its vectors have three components. Ambient coordinate count and subspace dimension are different quantities.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Do (1,0) and (2,0) form a basis of ℝ²?

Yes: both are nonzero.

No: they are dependent and span only a line.

Yes: there are two vectors.

Hint 1 · Find a starting point

A basis must be independent and span the space.

Hint 2 · Take the next step

The second vector is twice the first.

Show the reasoning

Answer: No: they are dependent and span only a line.

All combinations lie on the x-axis, so neither independence nor spanning ℝ² holds.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: coordinates depend on the basis

In $\mathbb R^2$ , let $\mathbf b_1=(1,1)^T$ and $\mathbf b_2=(1,-1)^T$ . To express $(3,1)^T$ , solve $c_1+c_2=3$ and $c_1-c_2=1$ . The basis coordinates are $(2,1)^T$ because $2\mathbf b_1+\mathbf b_2=(3,1)^T$ .

The vector itself has not changed; only its description has. Write a subscript such as $[\mathbf v]_B$ when multiple bases are in use.

Practice

Are $(1,2)^T$ and $(2,4)^T$ independent?
Does $(1,0,0)^T,(0,1,0)^T$ span $\mathbb R^3$ ?
Find the coordinates of $(0,2)^T$ in the basis above.

Show worked solutions

No: the second is twice the first.
No: every generated vector has third component zero. They form a basis only of the $xy$ -plane.
Solve $c_1+c_2=0$ , $c_1-c_2=2$ : coordinates $(1,-1)^T$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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