We-suppose-in-R-2-the-base-i-j-we-have-these-vectors-u-m-2-m-i-2mj-v-m-1-i-m-1-j-m-R-1-Determinate-m-for-which-the-system-u-v-is-linear-dependant-det-

Question Number 94915 by mathocean1 last updated on 21/May/20

We suppose in R^{2} the base (\vec{i}; \vec{j}) .

we have these vectors :

\vec{u} = (m^{2} - m) \vec{i} + 2 m \vec{j};

\vec{v} = (m - 1) \vec{i} + (m + 1) \vec{j} m \in R^{*}

1) Determinate m for which the system

(\vec{u}; \vec{v}) is linear dependant (\det (\vec{u}; \vec{v}) = 0)

Answered by mr W last updated on 21/May/20

\frac{m^{2} - m}{m - 1} = \frac{2 m}{m + 1}

\frac{m - 1}{m - 1} = \frac{2}{m + 1} = 1

m + 1 = 2

\Rightarrow m = 1

Answered by mathmax by abdo last updated on 22/May/20

u (m^{2} - m, 2 m) v (m - 1, m + 1)

\det (u, v) = 0 \Rightarrow | \begin{matrix} m^{2} - m m - 1 \\ 2 m m + 1 \end{matrix} | = 0 \Rightarrow

(m^{2} - m) (m + 1) - 2 m (m - 1) = 0 \Rightarrow

(m - 1) (m^{2} + m - 2 m) = 0 \Rightarrow (m - 1) (m^{2} - m) = 0 \Rightarrow

{(m - 1)}^{2} m = 0 \Rightarrow m = 1 or m = 0 .