Linear functional
Hello there! Let me explain what a linear functional is in a way that even a five-year-old can understand.
A linear functional is a special type of function that takes in a vector and returns a single number. You can think of a vector as a bunch of numbers written in a row, like this: (1, 2, 3).
Now, when we apply a linear functional to a vector, it looks at all the numbers in the vector and does some special math to combine them into a single number. This math always follows two rules:
1. If we multiply the vector by a number, the value of the linear functional also gets multiplied by the same number.
2. If we add two vectors together, the value of the linear functional applied to the sum is the same as if we had applied the linear functional to each vector separately, and then added the results together.
Here's an example: let's say we have a linear functional that takes in a vector of three numbers and returns the sum of those three numbers. So if we apply this linear functional to the vector (1, 2, 3), we get 1 + 2 + 3 = 6.
Now, let's say we multiply the vector by 2, so it becomes (2, 4, 6). If we apply the same linear functional to this new vector, we get 2 + 4 + 6 = 12, which is twice the value we got before.
And finally, if we add the original vector (1, 2, 3) to the new vector (2, 4, 6), we get (3, 6, 9). If we apply the same linear functional to this vector, we get 3 + 6 + 9 = 18, which is the same as if we had applied the linear functional separately to (1, 2, 3) and (2, 4, 6), and then added the results together.
So that's what a linear functional does - it takes in a vector and returns a number, and it always follows those two special rules I explained earlier. I hope that helps!
A linear functional is a special type of function that takes in a vector and returns a single number. You can think of a vector as a bunch of numbers written in a row, like this: (1, 2, 3).
Now, when we apply a linear functional to a vector, it looks at all the numbers in the vector and does some special math to combine them into a single number. This math always follows two rules:
1. If we multiply the vector by a number, the value of the linear functional also gets multiplied by the same number.
2. If we add two vectors together, the value of the linear functional applied to the sum is the same as if we had applied the linear functional to each vector separately, and then added the results together.
Here's an example: let's say we have a linear functional that takes in a vector of three numbers and returns the sum of those three numbers. So if we apply this linear functional to the vector (1, 2, 3), we get 1 + 2 + 3 = 6.
Now, let's say we multiply the vector by 2, so it becomes (2, 4, 6). If we apply the same linear functional to this new vector, we get 2 + 4 + 6 = 12, which is twice the value we got before.
And finally, if we add the original vector (1, 2, 3) to the new vector (2, 4, 6), we get (3, 6, 9). If we apply the same linear functional to this vector, we get 3 + 6 + 9 = 18, which is the same as if we had applied the linear functional separately to (1, 2, 3) and (2, 4, 6), and then added the results together.
So that's what a linear functional does - it takes in a vector and returns a number, and it always follows those two special rules I explained earlier. I hope that helps!
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