Matrices, an important function in both calculus and linear algebra, can be defined geometrically as a method of writing down the vectors of a coordinate system. So, by definition, a 2x2 matrix is the graph of a parallelogram, formed by the vectors of each column, drawn from the origin of a cartesian plane.
For example, say we have matrix A such that:
Then, we know that A can be drawn as a parallelogram with vectors:
and
drawn from the origin.
Matrices have countless uses; including quantum mechanics, engineering, and programming, and we can manipulate its elements so that the matrix can achieve a specific goal.
One of the most famous matrices is the Identity, or ‘unit’ matrix. Represented by the letter ‘I’, this function is similar to the number 1 in integer multiplication.
If you multiply any vector or matrix by the identity matrix, you get the value of the multiplicand.
But the identity matrix is not the only special matrix - we can create matrices for transformations, translations, reflections, or even rotations. If you multiply a point, vector, or another matrix by the rotation matrix,
the result will be a θ° counterclockwise rotation of your original image.
So, if you wanted a matrix A such that it rotates every image by 90° counter clockwise, you would plug in 90° for θ to get:
since cos(90) = 0 and sin(90) = 1.
While this alone is useful, what if there is something else that the rotation matrix can be used for?
If we take a matrix that rotates by (a+b) counterclockwise,
we know that we can break the rotation into two different matrices - a counterclockwise rotation by (a) and a counterclockwise rotation by (b), resulting in the following product:
Thus, we have proved the trigonometric identities:
cos(a+b) = cos a cos b - sin a sin b
and
sin(a+b) = sin a cos b+ cos a sin b
with matrices.
These are but a few of the uses for matrices, but there are so many more!
What is your favorite matrix/use of matrices?
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