Tensor products

1. Formula
   Observation

   Slide text: "V, W : vector spaces".

2. Audio
   Observation

   Speaker says: "We're going to start with some vector spaces, a vector space V, vector space W."

V

A vector space used as the domain of a linear map.

Vector space

1. Formula
   Observation

   Slide text: "V, W : vector spaces".

2. Audio
   Observation

   Speaker says: "a vector space V, vector space W."

W

A vector space used as the codomain of a linear map.

Vector space

1. Formula
   Observation

   Slide text: "f : $V \rightarrow W$ is linear if".

2. Audio
   Observation

   Speaker says: "And a function f from V to W is linear if it respects the structure of the objects..."

f

A function between vector spaces being tested for linearity.

Function from V to W

1. Formula
   Observation

   Slide formula: "$f(v_1 + v_2) = f(v_1) + f(v_2)$".

$v_1$

An arbitrary vector in V appearing in the additivity condition.

Element of V

1. Formula
   Observation

   Slide formula: "$f(v_1 + v_2) = f(v_1) + f(v_2)$".

$v_2$

An arbitrary vector in V appearing in the additivity condition.

Element of V

1. Formula
   Observation

   Slide formula: "$f(\alpha v) = \alpha f(v)$".

2. Audio
   Observation

   Speaker refers to "scalar multiplication".

1. The scalar field is not named on the slide or in the audio.

$\alpha$

A scalar used in the homogeneity condition for a linear map.

Scalar field of the vector spaces

1. Formula
   Observation

   Slide formula: "$f(\alpha v) = \alpha f(v)$".

v

An arbitrary vector in V used in the scalar-multiplication condition.

Element of V

1. Formula
   Observation

   Slide text: "U, V, W : vector spaces" and "f : $U \times V \rightarrow W$ is bilinear if".

2. Audio
   Observation

   Speaker says: "We're going to have three vector spaces, U, V, and W."

U

A vector space supplying the first input of a bilinear map.

Vector space

1. Formula
   Observation

   Slide text: "U, V, W : vector spaces" and formulas involving second-slot vectors v, $v_1$, $v_2$.

2. Audio
   Observation

   Speaker says: "a map f which takes a vector in U and a vector in V to a vector in W".

V

A vector space supplying the second input of a bilinear map; same letter as in the linear-map section but now in a different role.

Vector space

1. Formula
   Observation

   Slide text: "U, V, W : vector spaces" and "f : $U \times V \rightarrow W$ is bilinear if".

2. Audio
   Observation

   Speaker says: "to a vector in W".

W

The codomain vector space of a bilinear map.

Vector space

1. Formula
   Observation

   Slide text: "f : $U \times V \rightarrow W$ is bilinear if".

2. Audio
   Observation

   Speaker says: "And a map f which takes a vector in U and a vector in V to a vector in W is bilinear if it satisfies these properties."

f

A map from the Cartesian product $U \times V$ into W being tested for bilinearity.

Map $U \times V \rightarrow W$

1. Formula
   Observation

   Slide formula: "$f(u_1 + u_2, v) = f(u_1, v) + f(u_2, v)$".

$u_1$

An arbitrary vector in U appearing in the first-slot additivity condition.

Element of U

1. Caption evidence
   Observation

   Opening slide reads "Tensor Products", "Jim Fowler", "Spring 2009".

2. Audio
   Observation

   Speaker says: "Here's the seven minute introduction to tensor products."

3. Formula
   Observation

   Slide text: "We won't quite give an honest definition, but we'll learn how to calculate with $\otimes$."

The clip presents itself as a short introduction to tensor products. It explicitly states that it will not give a full formal definition, but will instead teach how to calculate with the tensor product symbol $\otimes$.

1. This is an introductory presentation rather than a complete formal development.

1. Formula
   Observation

   Slide text: "V, W : vector spaces" and "f : $V \rightarrow W$ is linear if" followed by "$f(v_1 + v_2) = f(v_1) + f(v_2)$" and "$f(\alpha v) = \alpha f(v)$".

2. Audio
   Observation

   Speaker says a function is linear "if it respects the structure of the objects, that is, f takes sums to sums and scalar multiplication to scalar multiplication."

A map f from one vector space V to another vector space W is linear exactly when it preserves vector addition and scalar multiplication. The displayed conditions are additivity and homogeneity.

1. V and W are vector spaces.

2. f : $V \rightarrow W$.

3. $v_1$, $v_2 \in V$.

4. $\alpha$ is a scalar.

5. $v \in V$.

1. V
2. W
3. f
4. $v_1$
5. $v_2$
6. $\alpha$
7. v

1. Formula
   Observation

   Slide text: "U, V, W : vector spaces" and "f : $U \times V \rightarrow W$ is bilinear if" followed by three displayed identities.

2. Audio
   Observation

   Speaker says: "Now a bilinear map is a somewhat more complicated object... And a map f which takes a vector in U and a vector in V to a vector in W is bilinear if it satisfies these properties."

A map f from the Cartesian product $U \times V$ into W is bilinear when it is additive and homogeneous separately in each input slot. The slide displays the corresponding identities for both slots.

1. U, V, W are vector spaces.

2. f : $U \times V \rightarrow W$.

3. u, $u_1$, $u_2 \in U$.

4. v, $v_1$, $v_2 \in V$.

5. $\alpha$ is a scalar.

1. U
2. V
3. W
4. f
5. $u_1$
6. $u_2$
7. v
8. u
9. $v_1$
10. $v_2$
11. $\alpha$

1. Formula
   Observation

   Slide text: "That is, $f(u, v)$ is linear in u if v is fixed" and "$f(u, v)$ is linear in v if u is fixed".

2. Audio
   Observation

   Speaker says: "all they're saying is that f of u, v is linear in the u coordinate if you hold v fixed, and it's linear in the v coordinate if you hold u fixed. So it's linear in both coordinates, but separately."

Bilinearity means linearity in each argument separately after the other argument is held fixed. This is the conceptual bridge from the algebraic identities to the later tensor-product characterization.

1. One input is fixed while checking linearity in the other input.

1. Definition of a bilinear map

1. Formula
   Observation

   Slide title: "Characterizing $U \otimes V$" and formula "\{ $\text{bilinear } U \times V \rightarrow W$ \} $\cong \text{Hom}(U \otimes V, W)$".

2. Formula
   Observation

   Slide text: "We'd like to build a vector space $U \otimes V$ so that bilinear maps $U \times V \rightarrow W$ are linear maps $U \otimes V \rightarrow W$."

3. Audio
   Observation

   Speaker says: "Well, U tensor V is a vector space so that bilinear maps from U and V to W are the same thing as linear maps from U tensor V into W."

1. The clip states the characterization but does not prove existence or uniqueness of $U \otimes V$.

The tensor product $U \otimes V$ is introduced as a vector space built so that bilinear maps out of $U \times V$ correspond to linear maps out of $U \otimes V$. The slide expresses this by identifying the collection of bilinear maps $U \times V \rightarrow W$ with Hom($U \otimes V$, W).

1. U, V, W are vector spaces.

2. The statement is presented as a characterization, not as a construction or proof.

1. Definition of a bilinear map
2. Definition of a linear map
3. $U \otimes V$
4. $\text{Hom}(U \otimes V, W)$
5. $\cong$

1. Formula
   Observation

   Slide title: Characterizing U ⊗ V.

2. Formula
   Observation

   Displayed equivalence: {bilinear $U \times V \to W$} ≅ Hom(U ⊗ V, W).

3. Audio
   Observation

   The speaker says they want to build a vector space U ⊗ V so that bilinear maps $U \times V \to W$ are linear maps U ⊗ $V \to W$.

The clip presents the tensor product as the vector space one wants to construct so that bilinear maps out of $U \times V$ become linear maps out of U ⊗ V. The displayed formula states this as an identification between the set of bilinear maps and Hom(U ⊗ V, W).

1. U, V, and W are treated as vector spaces.

2. The statement is presented as the design goal for U ⊗ V.

1. Formula
   Observation

   Slide text: Vectors in U ⊗ V are sums of u ⊗ v for $u \in U$, $v \in V$.

2. Audio
   Observation

   The speaker says a vector in U tensor V is just going to be a sum of little u's tensor little v's.

The video defines elements of the tensor product space operationally: every vector in U ⊗ V is a finite sum of simple tensors u ⊗ v with u from U and v from V.

1. $u \in U$

2. $v \in V$

1. Characterization goal for the tensor product

1. Formula
   Observation

   Displayed rule: u ⊗ v₁ + u ⊗ v₂ = u ⊗ (v₁ + v₂).

2. Audio
   Observation

   The speaker reads this rule aloud using vectors u, v₁, and v₂.

One of the special rules obeyed by symbols u ⊗ v is that adding two simple tensors with the same left factor equals tensoring that left factor with the sum of the right factors.

1. $u \in U$

2. v₁, v₂ ∈ V

1. Vectors in U ⊗ V are sums of simple tensors

1. Formula
   Observation

   Displayed rule: u₁ ⊗ $v + u$₂ ⊗ v = (u₁ + u₂) ⊗ v.

2. Audio
   Observation

   The speaker states the analogous rule for vectors u₁, u₂ in U and v in V.

The second special rule says that adding two simple tensors with the same right factor equals tensoring the sum of the left factors with that common right factor.

1. u₁, u₂ ∈ U

2. $v \in V$

1. Vectors in U ⊗ V are sums of simple tensors

1. Formula
   Observation

   Displayed rule: (αu) ⊗ $v = α$(u ⊗ v) = u ⊗ (αv).

2. Audio
   Observation

   The speaker explains that αu tensor v is the same as α times u tensor v, which is the same as u tensor αv.

The third rule shows compatibility with scalar multiplication: multiplying either input vector by a scalar before tensoring gives the same result as multiplying the whole simple tensor by that scalar, and the scalar may be placed on either side.

1. α is a scalar

2. $u \in U$

3. $v \in V$

1. Vectors in U ⊗ V are sums of simple tensors

1. Formula
   Observation

   Slide text: This is rigged so that a linear map U ⊗ $V \to W$ is the same as a bilinear map $U \times V \to W$.

2. Audio
   Observation

   The speaker says the rules look complicated but are rigged so that linear maps out of tensor products are exactly the same thing as bilinear maps.

The clip explains why the preceding algebraic rules are imposed: they are chosen so that a linear map out of U ⊗ V corresponds exactly to a bilinear map out of $U \times V$. In other words, the tensor rules mirror bilinearity.

1. Applies to maps into a common target space W.

1. Characterization goal for the tensor product
2. Tensor additivity in the second slot
3. Tensor additivity in the first slot
4. Scalar can move across the tensor product

1. Formula
   Observation

   Slide title: Bilinear versus Linear.

2. Formula
   Observation

   Displayed formula: For a bilinear map f : $U \times V \to W$, f(u, v₁ + v₂) = f(u, v₁) + f(u, v₂).

3. Audio
   Observation

   The speaker says that if we have a bilinear map f from U cross V into W, one of the properties it satisfies is this additivity in the second argument.

The video illustrates bilinearity by writing one of its defining properties: for fixed u, the map v ↦ $f(u, v)$ is additive, so f(u, v₁ + v₂) splits into f(u, v₁) + f(u, v₂).

1. f : $U \times V \to W$ is bilinear

2. $u \in U$

3. v₁, v₂ ∈ V

1. Purpose of the tensor-product rules

1. Formula
   Observation

   Slide text: "That is, $f(u, v)$ is linear in u if v is fixed" and "$f(u, v)$ is linear in v if u is fixed".

2. Audio
   Observation

   Speaker says bilinearity means "linear in both coordinates, but separately."

For a map f : $U \times V \rightarrow W$, the bilinearity conditions are equivalent to saying that f is linear in its first argument when the second argument is fixed, and linear in its second argument when the first argument is fixed.

1. U, V, W are vector spaces.

2. f : $U \times V \rightarrow W$.

For all fixed choices of one argument and all varying choices of the other argument.

1. Formula
   Observation

   Slide text: "Multilinear algebra—which initially appeared more complicated than linear algebra—is subsumed by linear algebra."

2. Audio
   Observation

   Speaker says: "we can take multilinear algebra... and put it inside linear algebra. That means that multilinear algebra becomes just a special case of linear algebra."

1. The clip does not spell out the precise categorical or universal-property formulation beyond the displayed correspondence.

If the vector space $U \otimes V$ can be built so that bilinear maps $U \times V \rightarrow W$ correspond to linear maps $U \otimes V \rightarrow W$, then multilinear algebra is absorbed into linear algebra as a special case.

1. The tensor product $U \otimes V$ exists with the stated characterizing property.

For the class of bilinear/multilinear constructions discussed in the clip.

1. Formula
   Observation

   Displayed formula: {bilinear $U \times V \to W$} ≅ Hom(U ⊗ V, W).

2. Audio
   Observation

   The speaker says they would like to build a vector space U ⊗ V so that bilinear maps $U \times V \to W$ are linear maps U ⊗ $V \to W$.

The tensor product U ⊗ V is characterized so that bilinear maps $U \times V \to W$ correspond to linear maps U ⊗ $V \to W$, expressed as {bilinear $U \times V \to W$} ≅ Hom(U ⊗ V, W).

1. U, V, and W are vector spaces.

2. The construction of U ⊗ V is assumed to satisfy the stated characterization.

For the displayed spaces U, V, and W.

1. Formula
   Observation

   Slide text: This is rigged so that a linear map U ⊗ $V \to W$ is the same as a bilinear map $U \times V \to W$.

2. Audio
   Observation

   The speaker says the rules for tensor products mirror the properties that bilinear maps have.

The algebraic rules imposed on symbols u ⊗ v are chosen precisely so that linear maps out of U ⊗ V match bilinear maps out of $U \times V$.

1. The previously listed tensor-product rules hold.

2. Maps are considered with common target W.

For the tensor-product rules shown in this clip.

1. Formula
   Observation

   The slide juxtaposes $f(u, v_1 + v_2) = f(u, v_1) + f(u, v_2)$ with f(u ⊗ ($v_1 + v_2$)) = f(u ⊗ $v_1$) + f(u ⊗ $v_2$).

2. Audio
   Observation

   The speaker says 'that looks exactly the same thing as the bilinear map up above.'

For the corresponding linear map f : U ⊗ $V \to W$, the additivity identity has the same formal shape as the bilinear additivity identity shown for f : $U \times V \to W$.

1. f : $U \times V \to W$ is bilinear

2. f : U ⊗ $V \to W$ is the corresponding linear map

3. $u \in U$ and $v_1$, $v_2 \in V$

For the displayed u, $v_1$, $v_2$ in the slide's setup.

1. Formula
   Observation

   The slide states 'ℝ^2 ⊗ ℝ^2 is spanned by' and lists four tensors.

2. Audio
   Observation

   The speaker says any sum of vectors in R2 tensor vectors in R2 can be rewritten as a sum of these four vectors using tensor-product properties.

1. Linear independence is not proved in this clip, although the speaker later calls them basis vectors.

If ℝ^2 = ⟨$e_1$, $e_2$⟩, then ℝ^2 ⊗ ℝ^2 is spanned by $e_1$ ⊗ $e_1$, $e_1$ ⊗ $e_2$, $e_2$ ⊗ $e_1$, and $e_2$ ⊗ $e_2$.

1. ℝ^2 = ⟨$e_1$, $e_2$⟩

For all elements of ℝ^2 ⊗ ℝ^2 described in the clip as sums of tensors from ℝ^2 × ℝ^2.

1. Formula
   Observation

   The warning box says 'You cannot combine ($e_1$ ⊗ $e_1$) + ($e_2$ ⊗ $e_2$).'

2. Audio
   Observation

   The speaker says 'not everything in R2 tensor R2 can be written as a vector in R2 tensor a vector in R2.'

1. The clip states this as a warning and verbal claim without proof.

Not every element of ℝ^2 ⊗ ℝ^2 can be written as a single elementary tensor x ⊗ y with x, y ∈ ℝ^2; some elements must be written as sums of elementary tensors.

1. Working in ℝ^2 ⊗ ℝ^2 with the basis-style spanning set introduced earlier

There exist elements of ℝ^2 ⊗ ℝ^2 that are not single pure tensors.

1. Audio
   Observation

   The speaker states that the properties of bilinear maps are mirrored in the properties of the tensor product, and that the tensor product is rigged so that maps out of tensor products are the same thing as bilinear maps.

The tensor product is constructed such that its algebraic properties mirror those of bilinear maps, making maps out of tensor products equivalent to bilinear maps.

No quantifiers are stated in the source.

1. Audio
   Observation

   The speaker gives an example: the tensor product of $R^3$ with $R^3$ is nine-dimensional, not six-dimensional.

The dimension of the tensor product of $R^3$ and $R^3$ is 9.

No quantifiers are stated in the source.

1. Formula
   Observation

   The slide shows dim($U \otimes V$) = dim Hom(U, V).

2. Audio
   Observation

   The speaker notes that the dimension of U tensor V is the same as the dimension of linear maps from U to V, and calls this a coincidence.

The dimension of the tensor product $U \otimes V$ is equal to the dimension of the space of linear maps Hom(U, V).

1. U and V are finite-dimensional vector spaces.

No quantifiers are stated in the source.

1. Audio
   Observation

   The speaker states that while U tensor V and linear maps from U to V have the same dimension, they are not naturally isomorphic.

The vector spaces $U \otimes V$ and Hom(U, V) are not naturally isomorphic.

No quantifiers are stated in the source.

1. Formula
   Observation

   Slide sequence: "\{ $\text{bilinear } U \times V \rightarrow W$ \} $\cong \text{Hom}(U \otimes V, W)$", then "We'd like to build a vector space $U \otimes V$ so that bilinear maps $U \times V \rightarrow W$ are linear maps $U \otimes V \rightarrow W$.", then "Multilinear algebra... is subsumed by linear algebra."

2. Audio
   Observation

   Speaker explains that $U \otimes V$ is a new vector space whose purpose is to turn bilinear maps into linear maps, and concludes that this places multilinear algebra inside linear algebra.

1. The derivation is motivational rather than a formal proof; existence and uniqueness of $U \otimes V$ are not established in the clip.

1. Expression

   $$$$
   Explanation

   Start with the notion of a bilinear map f : $U \times V \rightarrow W$.

   Justification

   This was defined earlier in the clip by separate linearity in each slot.

   Shown in the video
2. Expression

   $$\{ \text{bilinear } U \times V \rightarrow W \} \cong \text{Hom}(U \otimes V, W)$$
   Explanation

   Introduce a vector space $U \otimes V$ so that the collection of bilinear maps from $U \times V$ to W matches the collection of linear maps from $U \otimes V$ to W.

   Justification

   This is the displayed characterizing property of the tensor product.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Reinterpret bilinear data as linear data on a newly built space.

   Justification

   By the characterization, every bilinear map corresponds to a linear map out of $U \otimes V$.

   Derived from the video
4. Expression

   $$$$
   Explanation

   Conclude that multilinear algebra can be treated inside linear algebra.

   Justification

   The speaker states that once such $a U \otimes V$ is built, multilinear algebra becomes a special case of linear algebra.

   Shown in the video

The tensor product $U \otimes V$ is motivated as the device that converts bilinear maps into linear maps, thereby reducing multilinear questions to linear ones.

1. Formula
   Observation

   Displayed chain: f(u ⊗ (v₁ + v₂)) = f(u ⊗ v₁ + u ⊗ v₂) = f(u ⊗ v₁) + f(u ⊗ v₂).

2. Audio
   Observation

   The speaker says the tensor transforms in this way where u tensor that sum is u tensor v₁ plus u tensor v₂.

1. Expression

   $$f(u \otimes (v_1 + v_2))$$
   Explanation

   Start with the corresponding linear map applied to a simple tensor whose second factor is a sum.

   Justification

   This is the setup shown on the slide for the linear map f : U ⊗ $V \to W$.

   Shown in the video
2. Expression

   $$= f(u \otimes v_1 + u \otimes v_2)$$
   Explanation

   Rewrite the tensor inside f using the tensor-product rule for addition in the second slot.

   Justification

   Uses the displayed identity u ⊗ (v₁ + v₂) = u ⊗ v₁ + u ⊗ v₂.

   Shown in the video
3. Expression

   $$= f(u \otimes v_1) + f(u \otimes v_2)$$
   Explanation

   Apply linearity of f to split the sum inside the argument.

   Justification

   Uses that f is now a linear map on U ⊗ V.

   Shown in the video

The bilinear additivity property in the second variable is reproduced as ordinary linearity after passing to the tensor product.

1. Formula
   Observation

   The lower displayed chain is f(u ⊗ ($v_1 + v_2$)) = f(u ⊗ $v_1 + u$ ⊗ $v_2$) = f(u ⊗ $v_1$) + f(u ⊗ $v_2$).

2. Audio
   Observation

   The speaker explains that F is now linear and the result matches the bilinear formula above.

1. The first equality uses a tensor-product relation that is shown but not separately proved in this clip.

1. Expression

   $$f(u \otimes (v_1 + v_2))$$
   Explanation

   Start with the image under the linear map of the tensor u ⊗ ($v_1 + v_2$).

   Justification

   This is the left-hand side displayed on the slide.

   Shown in the video
2. Expression

   $$= f(u \otimes v_1 + u \otimes v_2)$$
   Explanation

   Rewrite the input tensor using distributivity in the second slot.

   Justification

   The slide displays the intermediate equality u ⊗ ($v_1 + v_2$) = u ⊗ $v_1 + u$ ⊗ $v_2$ inside f(...); the clip treats this as a property of tensor products.

   Shown in the video
3. Expression

   $$= f(u \otimes v_1) + f(u \otimes v_2)$$
   Explanation

   Apply linearity of f on U ⊗ V to the sum of two tensors.

   Justification

   The speaker explicitly says 'F is now linear' and the slide labels f : U ⊗ $V \to W$ as the corresponding linear map.

   Shown in the video

The displayed chain shows why the linear-map formula has the same additive form as the bilinear formula above it.

1. Formula
   Observation

   The slide displays ($e_1$ ⊗ $e_1$) + ($e_1$ ⊗ $e_2$) = $e_1$ ⊗ ($e_1 + e_2$).

2. Audio
   Observation

   The speaker says 'by using these kind of properties, any sum of tensors can be rewritten as a sum of those four tensors up above.'

1. Only one explicit rewrite is shown; the general reduction to the four listed tensors is asserted verbally rather than fully derived.

1. Expression

   $$(e_1 \otimes e_1) + (e_1 \otimes e_2)$$
   Explanation

   Begin with a sum of two elementary tensors that share the same left factor $e_1$.

   Justification

   This is the example written on the slide under 'Observe how these add'.

   Shown in the video
2. Expression

   $$= e_1 \otimes (e_1 + e_2)$$
   Explanation

   Factor out the common left tensor factor and combine the right factors into a sum.

   Justification

   This is the displayed equality, illustrating the tensor-product distributive property.

   Shown in the video
3. Expression

   $$\text{Repeat such rewrites to reduce arbitrary sums to the four listed tensors.}$$
   Explanation

   The speaker states that these properties allow any sum of tensors to be rewritten using the four spanning tensors.

   Justification

   This step is asserted in the audio, not shown as a full symbolic derivation in the clip.

   Shown in the video

The example demonstrates the mechanism behind the claim that ℝ^2 ⊗ ℝ^2 is spanned by the four listed elementary tensors.

1. Formula
   Observation

   The question box asks for ($e_1 + e_2$) ⊗ ($e_1 + e_2$), and the answer box gives ($e_1$ ⊗ $e_1$) + ($e_1$ ⊗ $e_2$) + ($e_2$ ⊗ $e_1$) + ($e_2$ ⊗ $e_2$).

2. Audio
   Observation

   The speaker says 'we can expand it out using the properties of tensor product' and reads the four-term answer.

1. The clip shows the start and end of the expansion but not the intermediate bilinear distribution steps in full symbolic detail.

1. Expression

   $$(e_1 + e_2) \otimes (e_1 + e_2)$$
   Explanation

   Start from the tensor product of the sum $e_1 + e_2$ with itself.

   Justification

   This is the question displayed in the red box.

   Shown in the video
2. Expression

   $$= e_1 \otimes (e_1 + e_2) + e_2 \otimes (e_1 + e_2)$$
   Explanation

   Distribute over the left sum.

   Justification

   This intermediate line is not shown on screen, but it is the standard tensor-product distributive step implied by the speaker's phrase 'expand it out using the properties of tensor product'.

   Derived from the video
3. Expression

   $$= (e_1 \otimes e_1) + (e_1 \otimes e_2) + (e_2 \otimes e_1) + (e_2 \otimes e_2)$$
   Explanation

   Distribute each term over the right sum to obtain the four elementary tensors.

   Justification

   This is the final expression displayed in the green Answer box and read aloud by the speaker.

   Shown in the video

The tensor ($e_1 + e_2$) ⊗ ($e_1 + e_2$) expands to the sum of all four basis-style elementary tensors listed earlier.

1. Formula
   Observation

   The slide adds 'For f bilinear, $f(a + b, a + b)$ is $f(a, a) + f(a, b) + f(b, a) + f(b, b)$.'

2. Audio
   Observation

   The speaker begins 'Because after all, if F were a...' before the clip ends.

1. The clip ends before the speaker completes the verbal explanation of the analogy.

1. Expression

   $$f(a + b, a + b)$$
   Explanation

   Consider a bilinear map evaluated on equal sums in both arguments.

   Justification

   This is the left-hand side displayed at the end of the clip.

   Shown in the video
2. Expression

   $$= f(a, a) + f(a, b) + f(b, a) + f(b, b)$$
   Explanation

   Expand by bilinearity in each argument to get four terms.

   Justification

   This is the exact formula shown on the slide; the clip does not display the intermediate distribution steps.

   Shown in the video

The final slide visually parallels the tensor-product expansion with the corresponding expansion of a bilinear map.

1. Formula
   Observation

   The slide visually connects the expansion of the bilinear map $f(a+b, a+b)$ to the expansion of the tensor product ($e_1 + e_2$) $\otimes (e_1 + e_2)$.