Is (6, 9) in the span of (2, 3) Step by Step

Step by step

The question

Question: is (6, 9) in the span of (2, 3)?

v₁ = [2; 3]

We call them v₁, and the vector we test b.

b = [6; 9]

The span of a vector is every linear combination of it: any number times it.

span: every c₁v₁

1 Write the vector equation

b is in the span if some numbers c₁ make this true.

We look for them.

c₁[2; 3] = [6; 9]

2 Write the augmented matrix

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

b goes after the bar.

= [2, 6; 3, 9]

We call the rows R 1 and R 2.

Careful!

Each vector is a column of the matrix, not a row.

Row 1 holds the first entries of v₁, and of b.

Careful: the vectors are the COLUMNS

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, 3; 3, 9]

→ [1, 3; 0, 0]

4 Read the answer

Each row is an equation again.

Row 1 says c₁ = 3.

c₁ = 3

So yes: b = 3v₁.

b is in the span.

b = 3v₁

5 Check

Let's check: we work out 3v₁.

= 3 × [2; 3]

= [3 × 2; 3 × 3]

= [6; 9]

We get (6, 9): that is b.

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 answer.

Last: check.

Answer