Question

Determine whether or not u&v are linearly dependent

1] u=(1,2,3,4),v=(4,3,2,1)

2]u=2t^2+4t-6, v=4t^2+8t-6

Answer 1

1)

$$A=\begin{bmatrix} 1 & 4 \\ 2 & 3 \\3 & 2 \\ 4 & 1 \end{bmatrix} \implies C_1\to 4C_1-C_2\implies \begin{bmatrix} 0 & 4 \\ 5 & 3 \\10 & 2 \\ 15 & 1 \end{bmatrix}$$

Thus, u and v are linearly dependent.

2)

 

$$A=\begin{bmatrix} 2 & 4 \\ 4 & 8 \\-6 & -6 \end{bmatrix} \implies\\ C_2\to C_2-2C_1\implies \begin{bmatrix} 2 & 0 \\ 4 & 0 \\-6 & 6 \\ \end{bmatrix}$$

Thus, u and v are linearly dependent.