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Span, Basis, Rank & Nullity

Useful preparation: Matrices, Elimination & Inverses

For a long equation, swipe sideways inside its formula box to see every term.

Meaning & notation

A span describes all outputs that a collection of directions can build. A basis removes redundant directions while retaining every possible output. Rank counts directions that survive a linear transformation, whereas nullity counts independent input directions that become invisible. These are dimensions of spaces, not simply counts of nonzero rows in the original display.

Subspace and span
A subspace contains zero and is closed under addition and scalar multiplication. The span of vectors is the set of all their linear combinations.
Basis and dimension
A basis is a linearly independent spanning list. Independence means that a zero linear combination forces every coefficient to be zero. Dimension is the number of vectors in a basis.
Rank and nullity
For A with n columns, rank is the dimension of its column space; nullity is the dimension of its nullspace, the set of x satisfying Ax=0.

Conditions before calculation

  • Use one fixed scalar field, here the real numbers. For a nonempty subset of a vector space, closure under all linear combinations proves it is a subspace.
  • Rank plus nullity equals the number of input coordinates n, not necessarily the number of output coordinates m.
  • Row operations preserve the nullspace and dependencies among columns. To obtain a basis for the original column space, select pivot columns from the original matrix.

Why the method works

1. Separate generation from uniqueness

Spanning asks whether every target can be generated. Independence asks whether its coordinates in the proposed basis are unique; a nonzero dependency allows different coordinate lists for the same vector.

2. Count inputs systematically

In echelon form, each pivot determines one variable and each free variable contributes one nullspace direction.

rank⁡(A)+nullity⁡(A)=n\operatorname{rank}(A)+\operatorname{nullity}(A)=n

3. Describe every solution

If one particular solution exists, adding a nullspace vector changes the input without changing the output. Conversely, subtracting any two solutions gives a nullspace vector.

{x:Ax=b}=xp+ker⁡A\{x:Ax=b\}=x_p+\ker A

Worked examples

EXAMPLE 1

Find rank and nullity for A with rows (1,2,0) and (0,0,1).

  1. Columns one and three are independent; column two is twice column one. Thus the image has dimension two.

  2. Solve the homogeneous equations and choose the free variable x 2=t\displaystyle x\, 2 = t.

    x1+2x2=0,x3=0x_1+2x_2=0,\quad x_3=0

  3. Write all solutions as multiples of one nonzero vector.

    x=t(−2,1,0),ker⁡A=span⁡{(−2,1,0)}x=t(-2,1,0),\quad \ker A=\operatorname{span}\{(-2,1,0)\}

Conclusion: Rank is 2, nullity is 1, and their sum is the three input coordinates.

EXAMPLE 2

Find a basis for the plane x+y+z=0\displaystyle x + y + z = 0 in real three-dimensional space.

  1. Set y=s\displaystyle y = s and z=t\displaystyle z = t because the equation then determines x.

  2. Separate the two independent choices.

    (x,y,z)=s(−1,1,0)+t(−1,0,1)(x,y,z)=s(-1,1,0)+t(-1,0,1)

  3. If this combination is zero, its second and third coordinates force s=t=0\displaystyle s = t = 0. Both vectors also satisfy the plane equation.

Conclusion: The displayed vectors form a basis; the plane has dimension two.

Pitfalls & counterexamples

Tempting claim: Any plane in three-dimensional space is a vector subspace.

Correction: A subspace must include the origin and remain closed under linear combinations.

Test the claim: The plane x+y+z=1\displaystyle x + y + z = 1 omits the zero vector and is not a subspace.

Check your understanding

Can four vectors be linearly independent in real three-dimensional space?

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Compare your reasoning

No.

A three-row matrix has at most three pivots, so four columns leave a free variable in the homogeneous system.

Connect this idea

Basis coordinates make abstract vector spaces computable. The same ideas apply to polynomials, functions, and solution spaces of homogeneous linear differential equations.

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Reference reading

Use these texts to study the underlying definitions and theorems in greater depth. The explanations and examples above are written for Numerive.

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