Common Core Standards: Math
11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
Students should feel pretty comfortable with matrices by the time you get to this standard. In particular, we hope they're feeling good about matrix multiplication, because the worlds of matrices and vectors are about to collide. Hopefully, they'll emerge vectorious. (Get it?)
Students should understand that a vector can be represented by a one-column matrix. Each number represents the components of the vector. (This can happen in 2D space or in 3D space.)
Let's say we have a vector with components <4, 1, 2> (meaning it moves 4 units along the x-axis, 1 unit along the y-axis, and 2 units along the z-axis). We can represent it as a matrix with one column:
Easy enough, yes?
Students should understand how to multiply a matrix by a vector. So let's take a matrix of appropriate dimensions and multiply it by our vector. (Notice the order in which we did that. The order matters.)
Remind your students and remind them again: we want to go across the rows of the matrix as we work our way down the vector.
To start with, we multiply 2 × 4 + 3 × 1 + 2 × 2 = 8 + 3 + 4 = 15.
For the second row, we have 1 × 4 + 5 × 1 + 3 × 2 = 4 + 5 + 6 = 15.
And finally, our last row gives us 0 × 4 + 3 × 1 + 6 × 2 = 0 + 3 + 12 = 15.
Our resulting vector is
Students should understand that multiplying a matrix and a vector is really just a combination of multiplication and addition—a way to transform that vector in 2D or 3D space. When we multiply a matrix and a vector together, we transform that vector in some way and change its components.