Find the Norm [[4-1i],[4-2i],[2+2i],[3-3i]]

Problem

‖[4−i],[4−2*i],[2+2*i],[3−3*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∥=√(,(∑_i=1^n)(|)*(z_i)|^2)

2. Calculate the squared magnitude |z|^2 for each component by summing the squares of the real and imaginary parts, using |a+b*i|^2=a^2+b^2

|4−i|^2=4^2+(−1)^2=16+1=17

|4−2*i|^2=4^2+(−2)^2=16+4=20

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

|3−3*i|^2=3^2+(−3)^2=9+9=18

3. Sum the squared magnitudes of all components.

17+20+8+18=63

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

√(,63)=√(,9⋅7)=3√(,7)

Final Answer

‖[4−i],[4−2*i],[2+2*i],[3−3*i]‖=3√(,7)

Want more problems? Check here!