SME
Linear Algebra · Problem type
Coordinates in a basis
Use this for “Find the coordinates of x = ⟨5, 1⟩ in the basis b₁ = ⟨1, 1⟩, b₂ = ⟨1, −1⟩.”
Diagram
c₁b₁ = ⟨3, 3⟩
|c₁b₁|
=
3
√
2
≈
4.243.
c₂b₂ = ⟨2, −2⟩
|c₂b₂|
=
2
√
2
≈
2.828.
Parallelogram: x = ⟨5, 1⟩
|x|
=
√
26
≈
5.099
at
11.31°.
This is an example. Type your own values to start a new problem.
c₁
⟨
b₁₁
,
b₁₂
⟩
+
c₂
⟨
b₂₁
,
b₂₂
⟩
=
⟨
x₁
,
x₂
⟩
D
Determinant of [b₁ b₂]
worked out (not typed)
Formulas
D
=
b₁₁
×
b₂₂
−
b₂₁
×
b₁₂
−2 = 1 × (−1) − 1 × 1
c₁
=
x₁
×
b₂₂
−
b₂₁
×
x₂
D
3
=
5 × (−1) − 1 × 1
−2
c₂
=
b₁₁
×
x₂
−
x₁
×
b₁₂
D
2
=
1 × 1 − 5 × 1
−2
x₁
=
c₁
×
b₁₁
+
c₂
×
b₂₁
5 = 3 × 1 + 2 × 1
x₂
=
c₁
×
b₁₂
+
c₂
×
b₂₂
1 = 3 × 1 + 2 × (−1)
Assumptions
•
The coordinates of x are the c₁ and c₂ with c₁b₁ + c₂b₂ = x: two equations in c₁ and c₂.
•
b₁ and b₂ are a basis of ℝ² exactly when D = det [b₁ b₂] ≠ 0; then the coordinates are unique.
•
Cramer’s rule solves the pair: put x in place of one column and divide by D.
Step-by-step
WE KNOW
b₁₁ = 1, b₁₂ = 1, b₂₁ = 1, b₂₂ = −1, x₁ = 5, x₂ = 1
FIND
determinant of [b₁ b₂] (D), coordinate along b₁ (c₁), coordinate along b₂ (c₂)
Step 1 · Find determinant of [b₁ b₂] (D)
Use D = b₁₁ × b₂₂ − b₂₁ × b₁₂
b₁
and
b₂
as
the
columns:
down
the
main
diagonal,
take
away
the
other
diagonal.
D
≠
0,
so
they
are
a
basis.
D
=
1
×
(−1)
−
1
×
1
D
=
−2
Step 2 · Find coordinate along b₁ (c₁)
Use c₁ = (x₁ × b₂₂ − b₂₁ × x₂) ÷ D
Cramer’s
rule:
put
x
in
place
of
b₁
(the
first
column),
take
that
determinant
and
divide
by
D
.
c₁
=
5 × (−1) − 1 × 1
−2
c₁
=
−5 − 1
−2
c₁
=
−
6
−2
c₁
=
3
Step 3 · Find coordinate along b₂ (c₂)
Use c₂ = (b₁₁ × x₂ − x₁ × b₁₂) ÷ D
Cramer’s
rule:
put
x
in
place
of
b₂
(the
second
column),
take
that
determinant
and
divide
by
D
.
c₂
=
1 × 1 − 5 × 1
−2
c₂
=
−
4
−2
=
2
Check: 3⟨1, 1⟩ + 2⟨1, −1⟩ = ⟨5, 1⟩
c₂
=
2
Check
Put the numbers back in. Both sides should match.
−2 = 1 × (−1) − 1 × 1 ✓
3
=
5 × (−1) − 1 × 1
−2
✓
2
=
1 × 1 − 5 × 1
−2
✓
5 = 3 × 1 + 2 × 1 ✓
1 = 3 × 1 + 2 × (−1) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.