Vector Spaces - The Space F^n - Cooperstein Exercise 12, page 10

[Math Amateur]

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ...

I am focused on Section 1.2 The Space $$\mathbb{F}^n$$ ...

I need help with Exercise 12 ... since I do not get the same answer as the author ...

Exercise 12 reads as follows:

View attachment 5103My attempt at a solution to this apparently simple problem is as follows:
Let $$\underline{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$$ ... ... We need to find $$v_1$$ and $$v_2$$ such that:

$$\begin{pmatrix} 2v_1 \\ 2v_2 \end{pmatrix} + \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$$ (mod 5) ... ... ... (1)Thus, given (1), we must have, for $$v_1$$ :

$$2 v_1 +3 = 1$$ (mod 5)

so we must have

$$2 v_1 = 1 - 3 = -2$$ (mod 5)

and so

$$v_1 = -1 \equiv 4$$ (mod 5)

----------------------------------------------

For $$v_2$$, given (1)we must have

$$2 v_2 + 4 = 3$$ (mod 5)

So then we must have

$$2 v_2 = 3 - 4 = -1 \equiv 4$$ (mod 5)

and so

$$v_2 = 2$$ Thus ...

$$\underline{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$$BUT ... in the "Hints to Selected Problems, Cooperstein gives the answer as$$\underline{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$$
So, my problem is ... why is there a discrepancy ... where is my error ...?

Hope someone can help ...

Peter

 

[Opalg]

It looks as though the error is Cooperstein's, not yours.