SME
Linear Algebra · Problem type
Are three vectors independent?
Use this for “Are ⟨1, 0, 1⟩, ⟨2, 1, 0⟩ and ⟨0, 1, 1⟩ linearly independent?”
Diagram
u = ⟨1, 0, 1⟩, with length |u| = 1.41.
v = ⟨2, 1, 0⟩, with length |v| = 2.24.
w = ⟨0, 1, 1⟩, with length |w| = 1.41.
u·(v × w) = 3, so the box’s volume is |3| = 3.
This is an example. Type your own values to start a new problem.
u
=
⟨
u₁
,
u₂
,
u₃
⟩
v
=
⟨
v₁
,
v₂
,
v₃
⟩
w
=
⟨
w₁
,
w₂
,
w₃
⟩
D
Determinant of [u v w]
worked out (not typed)
V
Volume of the box
worked out (not typed)
C
Independent or dependent
worked out (not typed)
Formulas
D
=
det
[[
u₁
,
v₁
,
w₁
],
[
u₂
,
v₂
,
w₂
],
[
u₃
,
v₃
,
w₃
]]
3 = det [[1, 2, 0], [0, 1, 1], [1, 0, 1]]
V
=
|
D
|
3 = |3|
C
=
independent
if
D
≠
0,
else
dependent
3 ≠ 0, so independent
Assumptions
•
Put u, v and w in the columns of a matrix: they are independent exactly when its det ≠ 0.
•
det = 0 means one vector is a combination of the others, and the box they span is flat.
•
|det| is the volume of the slanted box the three vectors span.
Step-by-step
WE KNOW
u₁ = 1, u₂ = 0, u₃ = 1, v₁ = 2, v₂ = 1, v₃ = 0, w₁ = 0, w₂ = 1, w₃ = 1
FIND
determinant of [u v w] (D), volume of the box (V), independent or dependent (C)
Step 1 · Find determinant of [u v w] (D)
Use D = det [[u₁, v₁, w₁], [u₂, v₂, w₂], [u₃, v₃, w₃]]
Write u, v and w as the columns, then expand along the first row with signs +, −, +.
D
=
det
[[1,
2,
0],
[0,
1,
1],
[1,
0,
1]]
det [[1, 2, 0], [0, 1, 1], [1, 0, 1]] = 1 × det [[1, 1], [0, 1]] − 2 × det [[0, 1], [1, 1]] + 0 × det [[0, 1], [1, 0]]
= 1 × (1 × 1 − 1 × 0) − 2 × (0 × 1 − 1 × 1) + 0 × (0 × 0 − 1 × 1)
= 1 × 1 − 2 × (−1) + 0 × (−1)
= 3
D
=
3
Step 2 · Find volume of the box (V)
Use V = |D|
The volume of the box is the size of the determinant (its sign only says which way the three vectors turn).
V
=
|3|
V
=
3
Step 3 · Find independent or dependent (C)
Use C = independent if D ≠ 0, else dependent
det ≠ 0: only c₁ = c₂ = c₃ = 0 gives c₁u + c₂v + c₃w = 0, so no vector is a combination of the others.
3 ≠ 0: independent
C
=
independent
if
3
≠
0,
else
dependent
C
=
independent
Check
Put the numbers back in. Both sides should match.
3 = det [[1, 2, 0], [0, 1, 1], [1, 0, 1]] ✓
3 = |3| ✓
3 ≠ 0, so independent ✓
Topic check
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