How do I choose a basis for the vectors in a small set?
Quick answer
First specify the target space. If you want a basis for the span of a given set, keep enough of its vectors to generate every vector in that set, but remove any vector that depends on the others. Then check both conditions: your chosen vectors span the target space, and the only way their linear combination equals zero is with all coefficients zero. A basis must satisfy both conditions [S1].
The question, the details, the next step.
I have a small list of vectors. What exactly am I trying to choose? Does “a basis for the vectors” mean a basis for the whole space they live in?
Start by naming the target space. Often the task is to find a basis for the span of the listed vectors—the set of all combinations you can make from them. That is not necessarily a basis for the whole surrounding space. A basis must both span its target space and be linearly independent; for example, two vectors that span a plane in R³ do not form a basis for all of R³ [S1].
A practical approach is to look for redundancy. If one vector can be made from the others, it may be removable without changing the span. The vectors you keep must still generate every original vector.
Can you show how to do that with actual numbers, including how to tell which vectors are redundant?
Suppose the set is
v₁ = (1, 0, 1), v₂ = (0, 1, 1), v₃ = (1, 1, 2), v₄ = (2, −1, 1).
Check the last two against the first two:
v₃ = v₁ + v₂, and v₄ = 2v₁ − v₂.
So both are redundant for generating this set’s span. Choose v₁ and v₂ as the candidate basis. This choice spans all four listed vectors: it includes v₁ and v₂, and the equations above show how to generate v₃ and v₄ from them.
For a small set, checking combinations directly can reveal which vectors are redundant. The important point is that the vectors you keep must still span the target.
How do I verify that the two vectors I kept are independent, rather than just assuming they are?
Set a linear combination equal to the zero vector and see whether only the trivial coefficients work:
a(1, 0, 1) + b(0, 1, 1) = (0, 0, 0).
Comparing coordinates gives a = 0 from the first coordinate and b = 0 from the second. The third coordinate, a + b = 0, is then satisfied too. Since there is no nonzero choice of a and b that makes the combination zero, v₁ and v₂ are independent. Together with the spanning check, this verifies they are a basis for the span of the four listed vectors [S1].
Notice the target matters: these two vectors span only a plane in R³, not all of R³. The check establishes a basis for their span, not for the entire ambient space.
What’s a reusable way to write up my answer so I don’t forget one of the checks?
State the target first, then give the candidate vectors and show both requirements. For this example, a concise write-up is:
“Let W = span{v₁, v₂, v₃, v₄}. Since v₃ = v₁ + v₂ and v₄ = 2v₁ − v₂, the vectors v₁ and v₂ span W. If av₁ + bv₂ = 0, comparing the first two coordinates gives a = b = 0, so v₁ and v₂ are linearly independent. Therefore {v₁, v₂} is a basis for W.”
Adapt the equations to your own vectors. If you are asked for a basis of a larger specified space, check that your selected vectors span that whole target—not merely the span of the original list. A basis requires both spanning and independence [S1].
A practical checklist
Write down the exact target space: the original vectors’ span or a larger specified space.
Look for vectors that are combinations of other vectors and remove redundant ones.
Show that the remaining vectors generate every vector in the target.
Set a general linear combination of the candidates equal to zero and solve for its coefficients.
Conclude they form a basis only if both spanning and independence are established.
A few more useful answers
Can two vectors be a basis for a plane in R³ but not for all of R³?
Yes. If they span the plane and are independent, they form a basis for that plane, but they do not span all of R³ [S1].
If I remove a vector, how do I know the span stays the same?
Check whether the vector you remove can be expressed as a combination of the vectors you keep. In the example, v₃ and v₄ are both expressible using v₁ and v₂, so the four vectors have the same span as v₁ and v₂.
Sources
References used for this guide.
- [S1] MIT OpenCourseWare: Independence, basis, and dimension (18.06SC, Fall 2011) ocw.mit.edu · Checked