Test subspace closure and describe spans, including polynomial examples.
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Try
Vectors u=(1,0,1) and v=(0,1,1) generate points (a,b,a+b) through au+bv. This is a plane through the origin. Adding a fixed offset changes it to a parallel affine plane, which usually no longer contains zero.
Predict
Predict which constraints define subspaces. A subspace W of a real vector space contains zero and is closed under addition and multiplication by every real scalar. Equivalently it is nonempty and closed under arbitrary linear combinations. The span of a list consists of all such combinations of its members.
Experiment
Generate three combinations of u,v, including one with a negative coefficient. Check z=x+y in each. Now test whether the shifted condition z=x+y+1 includes zero and whether adding two members preserves it. One failed closure condition is enough to refute a subspace claim.
Two different sets in R², with equal coordinate scales: y=x is a subspace line through zero; y=x+1 is its affine translate and misses zero. This is a two-dimensional comparison, separate from the plane example in R³.
Understand
Worked argument: every point satisfying z=x+y equals x(1,0,1)+y(0,1,1), so the two vectors span the entire solution plane. Conversely their arbitrary combination satisfies that equation. This proves equality of sets, not just that some sampled points lie on a plane.
Vectors need not be geometric arrows. Polynomials of degree at most two form a vector space under coefficientwise addition and scalar multiplication. The functions 1,t,t² span it. “Degree exactly two” is not a vector space: zero is missing, and leading terms can cancel.
Invalid subspace test: containing zero is sufficient. The union of the two coordinate axes contains zero, but (1,0)+(0,1) lies outside that union. Both closure conditions matter.
Apply
For each written proof, specify the ambient space and scalar field. Introduce arbitrary members rather than only convenient ones. For a counterexample, exhibit members whose sum or scalar multiple leaves the set and verify the original membership.
Continue
Depth: intersections of subspaces are subspaces, but unions generally are not. The span of their union is the smallest subspace containing both. This notion of generated structure also applies to polynomial, matrix and function spaces.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For u=(1,2,0), v=(0,1,3), enter the third coordinate of 2u−v.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
In p(t)=2(1+t)+3(t²), enter the coefficient of t.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which subset of R² is a subspace?
What is the span of an empty list of vectors?
Prove W={(x,y,z):x−2y+z=0} is a subspace of R³.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove the span of any finite list of vectors is a subspace.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
