Separate a dependence relation from a representation of a nonzero target.
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Try
The vectors (1,0),(0,1),(2,3) contain a redundant direction because the third is twice the first plus three times the second. Dependence means a nontrivial coefficient list produces zero; it does not mean the individual vectors are zero.
Predict
Predict why distinct nonzero vectors can still be dependent. A list v₁,…,vₖ is independent when Σaᵢvᵢ=0 forces every aᵢ=0. Place the vectors as matrix columns and solve the homogeneous coefficient system. A nonzero solution is an explicit certificate of dependence.
Experiment
Use the list (1,1),(2,2). Set the second coefficient to 1 and solve for the first. Then compare with (1,1),(1,−1), for which the coordinate equations force both coefficients zero. State the domain of coefficients: here they are real numbers, not only integers.
Understand
Worked proof: (1,0,1),(0,1,1) are independent. If a(1,0,1)+b(0,1,1)=0, the first coordinate yields a=0 and the second b=0. The third equation is then satisfied. Thus no nontrivial relation exists.
An independent list gives unique coefficients for any vector in its span. Subtract two representations of the same vector; the differences of their coefficients form a zero combination, so independence forces the representations to agree. Dependence instead allows one representation to be changed by adding a nonzero zero relation.
Invalid argument: no vector is a multiple of another, so the list is independent. That pairwise check misses relations involving three or more vectors, such as (1,0)+(0,1)−(1,1)=0.
Apply
In a dependence answer, supply actual coefficients and check their combination is zero. In an independence proof, show why every zero combination is trivial. These have different logical forms: an example proves dependence, but a few failed searches do not prove independence.
Continue
Depth: deleting a redundant vector leaves the span unchanged. Repeating this process extracts an independent spanning list. Over another scalar field, the same-looking coefficient equations can behave differently; the real-field assumption is part of this unit’s contract.
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Practice and reasoning
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For u=(1,0), v=(0,1), w=(2,3), the relation au+3v−w=0 requires what a?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Columns are u=(1,2), v=(2,4). In su+tv=0 with t=1, find s.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A list is independent when:
A list containing the zero vector is:
Prove (1,0,1),(0,1,1),(1,1,0) are independent over R.
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Prove that an independent list gives unique coefficients for each vector in its span.
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.
