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• Notes

Check which of the following sets form a vector space (show all details):
(a) $R^{2}=\left\{\left(x_{1}, x_{2}\right) | x_{i} \in R\right\}$ with standard addition and scalar multiplication.

Let

$\vec{u}=\left(u_{1}, u_{2}\right) \in R^{2}$
$\vec{v}=\left(v_{1}, v_{2}\right) \in R^{2}$
$\vec{w}=\left(w_{1}, w_{2}\right) \in R^{2}$

(1) $u+v=\left(u_{1}+v_{1}, u_{2}+v_{2}\right) \in R^{2}$

(2) $u+v=\left(u_{1}+v_{1}, u_{2}+v_{2}\right)=\left(v_{1}+u_{1}, v_{2}+u_{2}\right)=v+u$

(3) $(u+v)+w=\left(\left(u_{1}+v_{1})+w_{1}\right),\left(u_{2}+v_{2}\right)+w_{2}\right)$

$=\left(u_{1}+v_{1}+w_{1}, u_{2}+v_{2}+w_{2}\right)$

$=\left(u_{1}+\left(v_{1}+w_{1}\right), u_{2}+\left(v_{2}+w_{2}\right)=u+(v+w)\right.$

(4) $(0,0) \in R^{2} \Rightarrow 0+u=(0,0)+\left(u_{1}, u_{2}\right)=\left(u_{1}, u_{2}\right)=u$

$=u+0=\left(u_{1}, u_{2}\right)+(0,0)=\left(u_{1}, u_{2}\right)=u$

(5) $-u=\left(-u_{1},-u_{2}\right) \Rightarrow u+(-u)=\left(u_{1}, u_{2}\right)+\left(-u_{1},- u_{2}\right)$

$=\left(u_{1}-u_{1}, u_{2}-u_{2}\right)=(0,0)=0$

$=(-u+u) \Rightarrow\left(-u_{1}+u_{1},-u_{2}+u_{2}\right)=(0,0)=0$

(1) $K u=\left(K u_{1}, k u_{2}\right) \in R^{2}$

(2) $k(u+v)=k\left(u_{1}+v_{1}, u_{2}+v_{2}\right)=\left(k u_{1}+k v_{1}, K u_{2}+k v_{2}\right)$

$=\left(k u_{1}, k u_{2}\right)+\left(k v_{1}, k v_{2}\right)$

$=k u+k v$

(3) $(k+m)u=(k+m)\left(u_{1}, u_{2}\right)=\left((k+m) u_{1},(k+m) u_{2}\right)$

$=\left(k u_{1}+m u_{1}, k u_{2}+m u_{2}\right)$
$=\left(k u_{1}, k u_{2}\right)+\left(m u_{1}, m u_{2}\right)$

$=k u+m u$

(4) $k(mu)=k\left(m u_{1}, mu_{2}\right)=\left(k m u_{1}, k m u_{2}\right)=(k m)\left(u_{1}, u_{2}\right)$

$=(k m) u$

(5) $1 u=1\left(u_{1}, u_{2}\right)=\left(1 \cdot u_{1}, 1 \cdot u_{2}\right)=\left(u_{1}, u_{2}\right)=u$

(b) Let $V=R^{2}$ with standard addition of vectors, and scalar multiplication is
defined as:

$k u :=\left(k u_{1}, 0\right), \quad \forall k \in R, u \in R^{2}$

Let $u=\left(u_{1}, u_{2}\right)$

$1 u=1\left(u_{1}, u_{2}\right)=\left(u_{1}, 0\right) \neq\left(u_{1}, u_{2}\right)$

$\neq u$

$v$ is not a Vector space

(c) The set of all $2 \times 2$ Matrices, $M_{22}=\left\{\left[\begin{array}{ll}{a_{1}} & {a_{2}} \\ {a_{3}} & {a_{4}}\end{array}\right] | a_{i} \epsilon R\right\}$
with standard addition and scalarmultiplication on matrices.

Let $A=\left[\begin{array}{ll}{a_{1}} & {a_{2}} \\ {a_{3}} & {a_{4}}\end{array}\right], B=\left[\begin{array}{ll}{b_{1}} & {b_{2}} \\ {b_{3}} & {b_{4}}\end{array}\right], C=\left[\begin{array}{cc}{c_{1}} & {c_{2}} \\ {c_{3}} & {c_{4}}\end{array}\right]$