$($a$)$ Write out the $4-$dimensional vector for $(|0$:$)+|1$:$.$

$($b$)$ Let $B_1=\{|{}_1$:$),|{}_2$: and $B_2=\{|{}_1$:$),|{}_2$: be two orthonormal bases for $C^2.$ Prove that

$B_3=\{|{}_1$:$)ox|{}_1$:

is an orthonormal basis for $C^4.$ In other words, show that for each $|v$:$,$ and for

all pairs of distinct $|v$:$.$