Basic Vector Algebra

Vector Equality: given two vectors $u=⟨u_1,u_2,u_3,\dots ,u_n⟩$ and $v=⟨v_1,v_2,v_3,\dots ,v_n⟩$ in ${\mathbb{C}}^n$ or ${\mathbb{R}}^n$, the two vectors are equal if every component of $u$ equals every component of $v$.

Eg: Find real numbers $a$ and $b$ s.t.

$\left(\frac{a+b}{\sqrt{2}},\frac{a-b}{\sqrt{2}}\right)=(1,0)$

$\{\begin{array}{l}\frac{a+b}{\sqrt{2}}=1\\ \frac{a-b}{\sqrt{2}}=0\end{array}=\{\begin{array}{l}a=\frac{\sqrt{2}}{2}\\ b=\frac{\sqrt{2}}{2}\end{array}$

Complex conjugate: for every vector $v=⟨v_0,v_1,v_2,\dots ,v_n⟩$, there exists a conjugate denoted as $v^{\ast }$, given by

$v^{\ast }=⟨v_0^{\ast },v_1^{\ast },v_2^{\ast },\dots ,v_n^{\ast }⟩$

where $v_i^{\ast }$ is the complex conjugate of $v_i$ for $i=0,1,2,\dots ,n$.

Eg: given $v=⟨1,1+i⟩$, $v^{\ast }=⟨1,1-i⟩$

Vector Addition: given $u=⟨u_1,u_2,u_3,\dots ⟩$ and $v=⟨v_1,v_2,v_3,\dots ⟩$ in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$, their sum is:

$u+v=⟨u_1+v_1,u_2+v_2,\dots ,u_n+v_n⟩$

Properties:

• $u+v=v+u$ (commutativity)
• $u+(v+w)=(u+v)+w$ (associativity)
• $u+0=v$ (identity)

Geometric Representation

Parallelogram Rule   ·   Triangle Rule (both giving $u+v$).

Vector subtraction

The negative of a geometric vector $v$, denoted $-v$, is the vector with same magnitude and opposite direction.

$u-v=⟨u_1-v_1,u_2-v_2,\dots ,u_n-v_n⟩$

(Geometric picture: $⟨v_1,v_2⟩$ and its negative $⟨-v_1,-v_2⟩$.)

Scalar multiplication: $v=⟨v_1,v_2,v_3,\dots ,v_n⟩$ in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$, scalar $k$:

$kv=⟨k{v}_1,k{v}_2,k{v}_3,\dots ,k{v}_n⟩$

Linear combinations:

Given a vector $u$ and a set of $r$ vectors $⟨v_1,v_2,\dots ,v_r⟩$ in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$, then $u$ is a linear combination of $r$ if $u$ can be expressed as:

$u=k_1{v}_1+k_2{v}_2+k_3{v}_3+\cdots +k_r{v}_r={\sum }_{i=1}^r{k}_i{v}_i$