Coordinates of (2, 3, 4) in the basis (1, 0, 0), (1, 1, 0), (1, 1, 1) Step by Step

Step by step

The question

Question: find the coordinates of (2, 3, 4) in the basis (1, 0, 0), (1, 1, 0) and (1, 1, 1).

b₁ = [1; 0; 0], b₂ = [1; 1; 0], b₃ = [1; 1; 1]

We call the basis vectors b₁, b₂ and b₃, in this order, and the vector v.

v = [2; 3; 4]

The coordinates of v are the numbers that make v from the basis vectors, in the basis order.

coordinates: the numbers c₁, ..., c₃ in

v = c₁b₁ + c₂b₂ + c₃b₃

1 Write the vector equation

We look for c₁, c₂ and c₃.

c₁[1; 0; 0] + c₂[1; 1; 0] + c₃[1; 1; 1] = [2; 3; 4]

2 Write the augmented matrix

Each vector goes in as a column: the first entries make row 1, the second entries row 2.

v goes after the bar.

= [1, 1, 1, 2; 0, 1, 1, 3; 0, 0, 1, 4]

We call the rows R 1, R 2 and R 3.

3 Row reduce

We use the three row moves: swap two rows, times a row by a number that is not 0, add a multiple of one row to another.

They never change the answers.

Our goal: a leading 1 in each row, with 0s above and below it.

swap two rows

× a row by a number (not 0)

add a multiple of one row to another

→ [1, 0, 0, −1; 0, 1, 1, 3; 0, 0, 1, 4]

→ [1, 0, 0, −1; 0, 1, 0, −1; 0, 0, 1, 4]

4 Read the coordinates

Each row is an equation again.

Row 1: c₁ = −1.

c₁ = −1

Row 2: c₂ = −1.

c₂ = −1

Row 3: c₃ = 4.

c₃ = 4

We write them as a column, in the basis order: (−1, −1, 4).

[v]_B = [−1; −1; 4]

5 Check

Let's check: −b₁ − b₂ + 4b₃ must give v.

= −[1; 0; 0] − [1; 1; 0] + 4 × [1; 1; 1]

= −[1; 0; 0] − [1; 1; 0] + [4 × 1; 4 × 1; 4 × 1]

= −[1; 0; 0] − [1; 1; 0] + [4; 4; 4]

= [−1 − 1 + 4; −0 − 1 + 4; −0 − 0 + 4]

= [2; 3; 4]

We get (2, 3, 4): that is v.

So the answer is right!

Let's look back.

First: write the vector equation.

Then: write the augmented matrix.

Then: row reduce.

Then: read the coordinates.

Last: check.

Answer