Solve for lamda linear dependent function if lamda is given in question as expression

Question

Solve for lamda if vectors are linear dependent (1,2,3),(4,lamda,6),(4,6,9)

Answer 1

The position vectors of the three are <1,2,3>, <4,λ,6>, <4,6,9>, represented by vectors A, B, C.

For linear dependence: c₁A+c₂B+c₃C=0 for some scalars c₁, c₂, c₃. The zero vector 0 is <0,0,0>.

This gives us the equations represented:

R1: c₁+4c₂+4c₃=0

R2: 2c₁+λc₂+6c₃=0

R3: 3c₁+6c₂+9c₃=0⇒c₁+2c₂+3c₃=0

We need a non-trivial solution to these equations (at least one of the c's must be non-zero) for linear dependence.

Now let R2 become R2-2R1: (λ-8)c₂-2c₃=0; and R3 become R3-3R1: -6c₂-3c₃=0, making c₃=-2c₂.

Also  (λ-8)c₂-2c₃=0=(λ-8)c₂+4c₂⇒λ-8+4=0 (if c₂≠0), and λ=4.

COROLLARY

Let c₂=1 (arbitrarily), then:

R1: c₁+4+4c₃=0

R2: 2c₁+4+6c₃=0⇒c₁+2+3c₃=0

R3: c₁+2+3c₃=0=R2 (creates a redundancy in the set of vectors)

R1-R2: 2+c₃=0, c₃=-2⇒c₁=4⇒4A+B-2C=0:

4<1,2,3>+<4,4,6>-2<4,6,9>=<0,0,0>✔️

We've already seen that c₃=-2c₂, so if k=c₂, c₁=4k and 4k<1,2,3>+k<4,4,6>-2k<4,6,9>=<0,0,0> for all k≠0.