Find the Norm [[3+4i],[4+4i],[2+2i],[4-2i]]

Problem

‖[3+4*i],[4+4*i],[2+2*i],[4−2*i]‖

Solution

1. Identify the formula for the Euclidean norm (also known as the L^2 norm) of a complex vector v which is the square root of the sum of the squared magnitudes of its components.

∥v∥=√(,(∑_k=1^n)(|)*(z_k)|^2)

2. Calculate the squared magnitude |z|^2 for each component using the formula |a+b*i|^2=a^2+b^2

|3+4*i|^2=3^2+4^2=9+16=25

|4+4*i|^2=4^2+4^2=16+16=32

|2+2*i|^2=2^2+2^2=4+4=8

|4−2*i|^2=4^2+(−2)^2=16+4=20

3. Sum the squared magnitudes of all components.

(∑_^)(|)*(z_k)|^2=25+32+8+20

(∑_^)(|)*(z_k)|^2=85

4. Take the square root of the sum to find the final norm.

∥v∥=√(,85)

Final Answer

‖[3+4*i],[4+4*i],[2+2*i],[4−2*i]‖=√(,85)

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