Optional bridge
Bridge: matrix times vector
An optional next step: use dot products to multiply a 2×2 matrix by a vector.
This is an optional extra. It uses everything from the eight core lessons, especially the dot product. Skip it if you are not ready; nothing else depends on it.
The idea
A 2×2 matrix A can multiply a 2D vector v. The answer Av is another 2D vector. Each of its components is a dot product:
- the first component is row 1 of A dotted with v;
- the second component is row 2 of A dotted with v.
Worked example
A equals a 2 by 2 matrix, row 1: 2, 1; row 2: 0, 3 v equals the vector with components 4, 5
Find Av.
Dot each row with v
Av equals the vector with components 2 times 4 plus 1 times 5, and 0 times 4 plus 3 times 5Simplify
Av equals the vector with components 13, 15
Common mistakes
- Using columns of A instead of rows. Each answer component uses one row of A.
- Dropping a product. Each row gives two products to add.