Orthogonal Complements

1. Formula
   Observation

   Written as $\mathbb{R}^4$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'R four'.

$\mathbb{R}^4$

The vector space of all 4-dimensional real vectors.

Vector space

1. Formula
   Observation

   Written as $W$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'W'.

W

A subspace of $\mathbb{R}^4$ with basis $\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$.

Subspace of $\mathbb{R}^4$

1. Formula
   Observation

   Written as $W^\perp$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'W perp'.

$W^\perp$

The orthogonal complement of $W$, a subspace of $\mathbb{R}^4$ with basis $\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$.

Subspace of $\mathbb{R}^4$

1. Formula
   Observation

   The whiteboard shows the ambient space as $\mathbb{R}^4$ in the lines "Let's take $\mathbb{R}^4$" and "Now $W \oplus W^{\perp} = \mathbb{R}^4$".

$\mathbb{R}^4$

The four-dimensional real vector space used as the ambient space for the example.

Vector space over $\mathbb{R}$ with dimension 4.

1. Formula
   Observation

   The board states "Let $W$ have basis ..." and later uses $W$ in $W \oplus W^{\perp} = \mathbb{R}^4$.

W

A subspace of $\mathbb{R}^4$ whose displayed basis is $\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\right\}$.

Subspace of $\mathbb{R}^4$; in this example it is the span of the first two standard coordinate vectors.

1. Formula
   Observation

   The board states "Let $W^{\perp}$ have basis ..." and later uses $W^{\perp}$ in $W \oplus W^{\perp} = \mathbb{R}^4$.

$W^{\perp}$

The orthogonal complement of $W$ in $\mathbb{R}^4$, shown with basis $\left\{\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$.

Subspace of $\mathbb{R}^4$; here it is the span of the last two standard coordinate vectors.

1. Formula
   Observation

   The equation on the board reads $W \oplus W^{\perp} = \mathbb{R}^4$.

$\oplus$

Direct sum of subspaces, indicating that every vector in the ambient space is written as a sum of one vector from each subspace in a unique way.

Operation on subspaces of a vector space.

1. Formula
   Observation

   The board writes "a vector in $\mathbb{R}^4 = \begin{pmatrix}a\\b\\c\\d\end{pmatrix}$ can be written as ...".

$\begin{pmatrix}a\\b\\c\\d\end{pmatrix}$

A general vector in $\mathbb{R}^4$ with scalar coordinates $a,b,c,d$.

$a,b,c,d \in \mathbb{R}$.

1. Formula
   Observation

   The decomposition line shows the first summand as $\begin{pmatrix}a\\b\\0\\0\end{pmatrix}$.

$\begin{pmatrix}a\\b\\0\\0\end{pmatrix}$

The component of $(a,b,c,d)^T$ lying in $W$, obtained by keeping the first two coordinates and zeroing the last two.

Element of $W = \operatorname{span}\{(1,0,0,0)^T,(0,1,0,0)^T\}$.

1. Formula
   Observation

   The decomposition line shows the second summand as $\begin{pmatrix}0\\0\\c\\d\end{pmatrix}$.

$\begin{pmatrix}0\\0\\c\\d\end{pmatrix}$

The component of $(a,b,c,d)^T$ lying in $W^{\perp}$, obtained by zeroing the first two coordinates and keeping the last two.

Element of $W^{\perp} = \operatorname{span}\{(0,0,1,0)^T,(0,0,0,1)^T\}$.

1. Formula
   Observation

   Written as $W \oplus W^\perp = \mathbb{R}^4$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'W direct sum W perp, well, that's equal to R four'.

The direct sum of two subspaces $U$ and $V$, denoted $U \oplus V$, is the set of all vectors that can be written uniquely as the sum of a vector in $U$ and a vector in $V$. In this example, $W \oplus W^\perp = \mathbb{R}^4$.

1. $W$ and $W^\perp$ are subspaces of $\mathbb{R}^4$

2. $W \cap W^\perp = \{\mathbf{0}\}$

1. Subspace
2. Orthogonal Complement

1. Audio
   Observation

   Spoken as 'let's say that W is the subspace that has these two vectors as a basis'.

A subset of a vector space that is itself a vector space under the same operations. Here, $W$ is a 2-dimensional subspace of $\mathbb{R}^4$.

1. Closed under vector addition

2. Closed under scalar multiplication

3. Contains the zero vector

1. Formula
   Observation

   Written as $W^\perp$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'we will let W perp have basis...'.

For a subspace $W$ of an inner product space $V$, the orthogonal complement $W^\perp$ is the set of all vectors in $V$ that are orthogonal to every vector in $W$.

1. Defined within an inner product space

1. Subspace

1. Formula
   Observation

   The whiteboard heading is "Orthogonal Complements" and the worked setup gives bases for $W$ and $W^{\perp}$ inside $\mathbb{R}^4$.

2. Audio
   Observation

   The lecturer says this gives "a nice basic idea of orthogonal complements to work with."

The clip presents a specific example in $\mathbb{R}^4$: $W$ is spanned by the first two standard basis vectors, and $W^{\perp}$ is spanned by the last two. This illustrates how a subspace and its orthogonal complement split the ambient space into coordinate blocks.

1. The ambient space is $\mathbb{R}^4$.

2. The displayed bases are taken as given on the board.

1. Formula
   Observation

   The board explicitly writes $W \oplus W^{\perp} = \mathbb{R}^4$.

2. Formula
   Observation

   Below it, a general vector $(a,b,c,d)^T$ is rewritten as $(a,b,0,0)^T + (0,0,c,d)^T$.

Using the chosen subspaces, every vector in $\mathbb{R}^4$ is decomposed into one part in $W$ and one part in $W^{\perp}$. The first summand keeps the first two coordinates, and the second keeps the last two.

1. $a,b,c,d \in \mathbb{R}$.

2. $W$ and $W^{\perp}$ are the subspaces listed on the board.

1. Concrete orthogonal-complement pair in $\mathbb{R}^4$

1. Audio
   Observation

   The lecturer says, "of two individual subspaces, the way that I write it is unique. There's no other way of writing that."

2. Formula
   Observation

   The board has the handwritten note "This representation is UNIQUE." next to the decomposition.

For the displayed decomposition, the lecturer emphasizes that writing a vector as a sum of one element of $W$ and one element of $W^{\perp}$ is unique: there is no alternative pair of summands producing the same vector under this direct-sum setup.

1. Applies to the decomposition associated with $W \oplus W^{\perp} = \mathbb{R}^4$ shown on the board.

1. Direct-sum decomposition of a vector in $\mathbb{R}^4$

1. Formula
   Observation

   The board states "Now $W \oplus W^{\perp} = \mathbb{R}^4$".

For the subspaces displayed on the board, $W \oplus W^{\perp} = \mathbb{R}^4$.

1. $W$ has basis $\left\{(1,0,0,0)^T,(0,1,0,0)^T\right\}$.

2. $W^{\perp}$ has basis $\left\{(0,0,1,0)^T,(0,0,0,1)^T\right\}$.

Universal over all vectors of $\mathbb{R}^4$ once the two subspaces are fixed.

1. Audio
   Observation

   The lecturer says the way of writing the vector using the two subspaces is unique and that there is no other way of writing it.

2. Formula
   Observation

   The board note reads "This representation is UNIQUE."

In the shown decomposition of a vector in $\mathbb{R}^4$ into a $W$-part plus a $W^{\perp}$-part, the representation is unique.

1. The vector is written using the specific subspaces $W$ and $W^{\perp}$ listed on the board.

For each vector in $\mathbb{R}^4$, there is exactly one such decomposition relative to the displayed $W$ and $W^{\perp}$.

1. Formula
   Observation

   Written as $\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix}$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'it can be written as a vector from this subspace plus a vector from this subspace'.

1. Expression

   $$\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix}$$
   Explanation

   Start with an arbitrary vector in $\mathbb{R}^4$.

   Justification

   Given in the problem statement.

   Shown in the video
2. Expression

   $$= \begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix}$$
   Explanation

   Split the vector into a part in $W$ (first two components) and a part in $W^\perp$ (last two components).

   Justification

   Definition of direct sum and the specific bases chosen for $W$ and $W^\perp$.

   Shown in the video
3. Expression

   $$\begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix} \in W$$
   Explanation

   The first vector is a linear combination of the basis vectors of $W$: $4\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} + 3\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$.

   Justification

   Definition of span/basis.

   Derived from the video
4. Expression

   $$\begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix} \in W^\perp$$
   Explanation

   The second vector is a linear combination of the basis vectors of $W^\perp$: $-2\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} + 6\begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$.

   Justification

   Definition of span/basis.

   Derived from the video

Any vector in $\mathbb{R}^4$ can be uniquely decomposed into a sum of a vector in $W$ and a vector in $W^\perp$.

1. Formula
   Observation

   The board writes a general vector $(a,b,c,d)^T$ and then rewrites it as $(a,b,0,0)^T + (0,0,c,d)^T$.

1. Expression

   $$\begin{pmatrix}a\\b\\c\\d\end{pmatrix}$$
   Explanation

   Start with an arbitrary vector in $\mathbb{R}^4$.

   Justification

   Given directly on the board as the object to decompose.

   Shown in the video
2. Expression

   $$\begin{pmatrix}a\\b\\0\\0\end{pmatrix} + \begin{pmatrix}0\\0\\c\\d\end{pmatrix}$$
   Explanation

   Separate the first two coordinates from the last two coordinates.

   Justification

   Coordinate-wise addition reproduces the original vector exactly.

   Shown in the video
3. Expression

   $$\begin{pmatrix}a\\b\\0\\0\end{pmatrix} \in W,\quad \begin{pmatrix}0\\0\\c\\d\end{pmatrix} \in W^{\perp}$$
   Explanation

   Identify the first summand as lying in $W$ and the second as lying in $W^{\perp}$ using the bases shown above.

   Justification

   Each summand is a linear combination of the corresponding displayed basis vectors.

   Derived from the video

The example visually demonstrates the decomposition $\mathbb{R}^4 = W \oplus W^{\perp}$ by splitting coordinates into two orthogonal blocks.

1. Formula
   Observation

   Written as $\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix}$ on the whiteboard.

2. Audio
   Observation

   Spoken as 'let's say for example... some random vector 4, 3, minus 2, 6'.

Write the vector $\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix}$ as a sum of a vector in $W$ and a vector in $W^\perp$.

1. $W = \text{span}\left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \right\}$

2. $W^\perp = \text{span}\left\{ \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \right\}$

3. Target vector: $\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix}$

Find $w \in W$ and $v \in W^\perp$ such that $w + v = \begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix}$.

1. Expression

   $$\begin{pmatrix} 4 \\ 3 \\ -2 \\ 6 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix}$$
   Explanation

   Separate the components corresponding to the bases of $W$ and $W^\perp$.

   Justification

   Direct sum definition and chosen bases.

   Shown in the video

$w = \begin{pmatrix} 4 \\ 3 \\ 0 \\ 0 \end{pmatrix}$, $v = \begin{pmatrix} 0 \\ 0 \\ -2 \\ 6 \end{pmatrix}$

Check that $w \in W$, $v \in W^\perp$, and $w+v$ equals the original vector.

1. Formula
   Observation

   The full worked board content gives bases for $W$ and $W^{\perp}$, states $W \oplus W^{\perp} = \mathbb{R}^4$, and decomposes $(a,b,c,d)^T$.

2. Audio
   Observation

   The lecturer closes by saying this gives a basic idea of orthogonal complements.

Show how a general vector in $\mathbb{R}^4$ is written as a sum of one vector from $W$ and one vector from $W^{\perp}$ for the subspaces listed on the board.

1. Ambient space: $\mathbb{R}^4$.

2. Basis of $W$: $\left\{(1,0,0,0)^T,(0,1,0,0)^T\right\}$.

3. Basis of $W^{\perp}$: $\left\{(0,0,1,0)^T,(0,0,0,1)^T\right\}$.

4. General vector: $(a,b,c,d)^T$.

Produce the explicit $W \oplus W^{\perp}$ decomposition and state whether it is unique.

1. Expression

   $$\begin{pmatrix}a\\b\\c\\d\end{pmatrix} = \begin{pmatrix}a\\b\\0\\0\end{pmatrix} + \begin{pmatrix}0\\0\\c\\d\end{pmatrix}$$
   Explanation

   Split the vector into first-two-coordinate and last-two-coordinate parts.

   Justification

   Direct coordinate addition.

   Shown in the video
2. Expression

   $$\begin{pmatrix}a\\b\\0\\0\end{pmatrix} \in W,\quad \begin{pmatrix}0\\0\\c\\d\end{pmatrix} \in W^{\perp}$$
   Explanation

   Recognize the two summands as belonging to the displayed subspaces.

   Justification

   They are linear combinations of the respective bases written on the board.

   Derived from the video
3. Expression

   $$$$
   Explanation

   Conclude that the representation is unique.

   Justification

   Stated explicitly by the lecturer and written on the board as "This representation is UNIQUE."

   Shown in the video

$(a,b,c,d)^T = (a,b,0,0)^T + (0,0,c,d)^T$, with the first term in $W$, the second in $W^{\perp}$, and the decomposition unique.

Adding the two displayed summands coordinatewise recovers $(a,b,c,d)^T$; membership follows from the bases shown on the board.

1. Animation
   Observation

   Instructor writes the basis vectors for $W$ and $W^\perp$ on the whiteboard.

1. Whiteboard

2. Marker

3. Basis vectors for $W$

4. Basis vectors for $W^\perp$

1. Vectors are written sequentially on the board.

1. The mathematical meaning of the vectors remains constant once written.

Establishes the specific subspaces $W$ and $W^\perp$ being used in the example.

1. Animation
   Observation

   Instructor writes the equation $W \oplus W^\perp = \mathbb{R}^4$ and then the vector decomposition.

1. Whiteboard

2. Marker

3. Equation $W \oplus W^\perp = \mathbb{R}^4$

4. Vector decomposition equation

1. Equations are written to show the relationship between the spaces and the specific vector split.

1. The underlying mathematical truth of the direct sum property.

Visually demonstrates how an arbitrary vector in $\mathbb{R}^4$ is broken down into components belonging to $W$ and $W^\perp$.

1. Diagram
   Observation

   The screen remains on a static whiteboard layout with the title "Orthogonal Complements", the subspace bases, the equation $W \oplus W^{\perp} = \mathbb{R}^4$, the decomposition line, and the note "This representation is UNIQUE."

2. Animation
   Observation

   No new writing or diagrammatic transformation occurs during the clip; only the lecturer gestures in the webcam window.

1. Title "Orthogonal Complements".

2. Lines defining bases for $W$ and $W^{\perp}$.

3. Equation $W \oplus W^{\perp} = \mathbb{R}^4$.

4. Decomposition of $(a,b,c,d)^T$.

5. Handwritten uniqueness note.

1. No mathematical content is added or erased during the 23-second excerpt.

1. The displayed subspaces, equation, and decomposition remain fixed throughout.

2. The uniqueness statement remains visible beside the decomposition.

The visual functions as a summary board: it fixes the example, states the direct-sum identity, and highlights the uniqueness conclusion while the lecturer verbally wraps up.

1. Audio
   Observation

   The lecturer explicitly counters the idea of alternate decompositions by saying, "There's no other way of writing that."

2. Formula
   Observation

   The board note says "This representation is UNIQUE."

One might think a vector could be split into a $W$-part and a $W^{\perp}$-part in multiple different ways.

In the displayed $W \oplus W^{\perp}$ setting, the decomposition is unique; the lecturer stresses that there is no other way to write the same vector using these two subspaces.

1. Audio
   Observation

   Spoken as 'that's what the direct sum means'.

The concept of direct sum is applied to the specific subspaces $W$ and $W^\perp$ to show they span $\mathbb{R}^4$.

1. Formula
   Observation

   $W^\perp$ is defined relative to $W$.

The orthogonal complement is a specific type of subspace related to another subspace via orthogonality.

1. Formula
   Observation

   The bases for $W$ and $W^{\perp}$ are written immediately above the equation $W \oplus W^{\perp} = \mathbb{R}^4$ and the coordinate split.

The concrete choice of bases for $W$ and $W^{\perp}$ is what makes the displayed coordinate decomposition possible.

1. Audio
   Observation

   The uniqueness remark is spoken directly about the just-written decomposition.

2. Formula
   Observation

   The note "This representation is UNIQUE." is placed next to the decomposition line.

The uniqueness claim is about the specific representation produced by the direct-sum decomposition shown on the board.

1. Audio
   Observation

   Spoken as 'what's nice about this representation... is unique'.

1. Direct Sum of Subspaces

1. Formula
   Observation

   Vector decomposition is shown on the board.

1. Direct Sum of Subspaces
2. Decomposing a Vector in $R^4$

1. Formula
   Observation

   The board contains a fully written example with $W$, $W^{\perp}$, and a vector decomposition in $\mathbb{R}^4$.

1. Concrete orthogonal-complement pair in $\mathbb{R}^4$
2. Direct-sum decomposition of a vector in $\mathbb{R}^4$

1. Audio
   Observation

   The lecturer says the way of writing the vector using the two subspaces is unique.

2. Formula
   Observation

   The board note reads "This representation is UNIQUE."

1. Uniqueness of the $W \oplus W^{\perp}$ representation

1. Formula
   Observation

   The decomposition line explicitly rewrites $(a,b,c,d)^T$ as $(a,b,0,0)^T + (0,0,c,d)^T$.

1. Direct-sum decomposition of a vector in $\mathbb{R}^4$
2. Worked example: decomposing a vector in $\mathbb{R}^4$ using $W$ and $W^{\perp}$