Show that the dot product is linear: Bra-ket notation

[lausco]

Homework Statement

Show that the dot product in two-dimensional space is linear:
<u|(|v> + |w>) = <u|v> + <u|w>

The Attempt at a Solution

I feel like I'm missing some grasp of the concept here ...
I would think to just distribute the <u| and be done in that one step,
but I'm being asked to prove this.
Is there a reason the <u| can't simply be distributed?

 

[collinsmark]

Homework Statement

Show that the dot product in two-dimensional space is linear:
<u|(|v> + |w>) = <u|v> + <u|w>

The Attempt at a Solution

I feel like I'm missing some grasp of the concept here ...
I would think to just distribute the <u| and be done in that one step,
but I'm being asked to prove this.
Is there a reason the <u| can't simply be distributed?

I think you're being asked to demonstrate this longhand, in order to prove it.

In other words, recall (for two dimensional space),

[tex] | v \rangle =
\begin{pmatrix}
v_1 \\
v_2
\end{pmatrix}, \ \
| w \rangle =
\begin{pmatrix}
w_1 \\
w_2
\end{pmatrix}, \ \
\langle u | = (u_1^*, u_2^*) [/tex]
and then work things out longhand, using more conventional methods, to eventually show that it does distribute.

 

[lausco]

I know that <u|v> = the length of |u> times the projection of |v> along |u> . . . Are the conjugates related the the projection?

 

[collinsmark]

I know that <u|v> = the length of |u> times the projection of |v> along |u> . . . Are the conjugates related the the projection?

I think so, yes. I'm not the best person to be explaining math, so don't rely on me for a graceful explanation of this. But yes, I think the conjugates ultimately comes down to some sort of generalization of projections.

[itex] | A \rangle [/itex] and [itex] \langle A | [/itex] represent conjugate transposes of one another. Of course if you deal only with real numbers, you don't need to worry about the conjugate. But in general, when dealing the complex numbers, the conjugate is necessary.

Perhaps it's easiest to demonstrate the motivation of this with a special case of taking the inner product of a vector with itself. In other words, let's examine [itex] \langle A | A \rangle [/itex].

Let's further simplify this to 1 dimension for now. Suppose we have a simple, one-dimensional vector [itex] | A \rangle = \left( 3 + i4 \right) [/itex]. Suppose our goal is to find the magnitude squared of this complex, one dimensional vector. In complete agreement with the length of A times the projection of A onto itself, we can find the length squared of A by finding [itex] \langle A | A \rangle [/itex]. In this example, the answer is (3 - i4)(3 + i4) = 25. Notice I found |A|² by multiplying the complex conjugate of A by A. In other words, |A|² = A^*A for this one dimensional case. Notice that since we were trying to find the magnitude squared of a vector, the answer will always be real, even if A is complex. We couldn't do that without the complex conjugate.

We can move on to larger dimensional spaces by saying that in general, [itex] \langle A | [/itex] is the conjugate transpose of [itex] | A \rangle [/itex].

 

[HalfLostAndSearching]

As a scientist, it is important to approach problems with a rigorous and logical mindset. In this case, we are asked to prove the linearity of the dot product in two-dimensional space using bra-ket notation. Let's start by reviewing the definition of the dot product in bra-ket notation: <u|v> = u1*v1 + u2*v2. This represents the dot product of two vectors u and v, where u1 and u2 are the components of vector u, and v1 and v2 are the components of vector v.

To prove the linearity of the dot product, we need to show that it follows the properties of linearity, which are additivity and homogeneity. Let's break down the given equation into its components to understand how it relates to these properties.

<u|(|v> + |w>) = <u|v> + <u|w>

First, let's look at the left side of the equation. The expression <u|(|v> + |w>) represents the dot product of vector u with the sum of vectors v and w. In other words, it is the dot product of vector u with the resultant vector obtained by adding vectors v and w. This follows the property of additivity, as the dot product of a vector with the sum of two other vectors is equal to the sum of the dot products of the vector with each individual vector.

Next, let's look at the right side of the equation. The expression <u|v> + <u|w> represents the sum of the dot products of vector u with vectors v and w. This also follows the property of additivity, as the sum of two dot products is equal to the dot product of the sum of the vectors.

Therefore, we can see that the given equation satisfies the property of additivity, which is one of the key properties of linearity. This proves that the dot product in two-dimensional space is linear in bra-ket notation.

To address your question about distributing the <u|, it is important to note that bra-ket notation follows certain rules and properties that may not align with the rules of traditional algebra. In this case, we are dealing with vectors and their components, so we cannot simply distribute the <u| as we would with regular numbers. Instead, we must use the definition of the dot product in bra-ket notation to prove its linearity.

In conclusion, the dot product