Expand the set of vectors in Exercise 2.7 to include a fourth vector, v 4 = (8, 5). Reformulate Exercise 2.8 to include the fourth vector by including v4 in V and an additional coefficient in x. Is...

Expand the set of vectors in Exercise 2.7 to include a fourth vector, v₄
= (8, 5). Reformulate Exercise 2.8 to include the fourth vector by including v4 in V and an additional coefficient in x. Is this system of equations consistent? Is the solution unique? Find a solution. If solutions are not unique, find another solution.

Exercise 2.8

The three vectors in Exercise 2.7 are linearly dependent. Find the linear function of v₁
and v₂
that equals v₃. Set the problem up as a system of linear equations to be solved. Let V = (v₁
v₂), and let x = (x₁
x₂) be the vector of unknown coefficients. Then, V_x
= v₃
is the system of equations to be solved for x.

(a) Show that the system of equations is consistent.

(b) Show that there is a unique solution.

(c) Find the solution.

Exercise 2.7

Plot the following vectors on a two-dimensional coordinate system.

By inspection of the plot, which pairs of vectors appear to be orthogonal? Verify numerically that they are orthogonal and that all other pairs in this set are not orthogonal. Explain from the geometry of the plot how you know there is a linear dependency among the three vectors.