Problem Name

Problem Description

Notes

linear_algebra_1

Use matrix multiplication to show that is the solution to the following system of equations. Hint: Show that the product of the coefficient matrix and the solution vector is equal to the right-hand-side vector.

linear_algebra_2

Find a point with on the intersection line of the planes and . Find the point with . Find a third point halfway between.

linear_algebra_3

Reduce this system to upper triangular form by two row operations:

 

Solve by back substitution for

linear_algebra_4

For any unit vectors and , find the dot products (actual numbers) of:
(a)
(b)
(c)

linear_algebra_5

If a 4 by 4 matrix has , find and and and .

linear_algebra_6

Prove that the square of a Markov matrix is also a Markov matrix.

linear_algebra_7

From and , find the components of and .

linear_algebra_8

Multiply these matrices:

linear_algebra_9

Add to and compare with

linear_algebra_10

Show that is not invertible.