Composing 3x3 matrices | Matrices | Precalculus | Khan Academy

1. Formula
   Observation

   The white matrix label A is shown with entries [[-3, 0, 0], [-3, -2, -4], [3, 3, 1]].

2. Audio
   Observation

   The introduction treats the displayed matrices as transformations in three dimensions.

A

A 3×3 matrix representing a transformation in three-dimensional space.

3×3 real matrices

1. Formula
   Observation

   The red matrix label B is shown with entries [[-2, 0, -2], [-3, 2, 4], [2, 3, -4]].

2. Audio
   Observation

   The second displayed matrix is identified as B.

B

A 3×3 matrix whose columns are transformed by A to form the columns of A∘B.

3×3 real matrices

1. Formula
   Observation

   The lower-left expression reads A ∘ B = [[6, _, 6], [4, _, 14], [-13, _, 2]], with the middle column blank.

2. Audio
   Observation

   The task concerns the composition and its unfilled middle column.

A \circ B

The composition transformation obtained by applying B first and then A; represented here as a 3×3 matrix.

3×3 real matrices

1. Formula
   Observation

   The vector [1, 0, 0]^T is written as part of the decomposition of the middle column of B.

2. Audio
   Observation

   The decomposition uses the coordinate direction along the x-axis.

\begin{bmatrix}1\\0\\0\end{bmatrix}

The standard basis vector in the x-direction in R^3.

R^3

1. Formula
   Observation

   The vector [0, 1, 0]^T appears multiplied by 2 in the decomposition of the middle column of B.

2. Audio
   Observation

   The decomposition also uses the coordinate direction along the y-axis.

\begin{bmatrix}0\\1\\0\end{bmatrix}

The standard basis vector in the y-direction in R^3.

R^3

1. Formula
   Observation

   The vector [0, 0, 1]^T appears multiplied by 3 in the decomposition of the middle column of B.

2. Audio
   Observation

   The last basis term uses the coordinate direction along the z-axis.

\begin{bmatrix}0\\0\\1\end{bmatrix}

The standard basis vector in the z-direction in R^3.

R^3

1. Formula
   Observation

   The matrix A is written as a 3x3 matrix with columns [-3, -3, 3], [0, -2, 3], and [0, -4, 1].

A

A 3x3 matrix representing a linear transformation.

3x3 real matrices

1. Formula
   Observation

   The matrix B is written as a 3x3 matrix with columns [-2, -3, 2], [0, 2, 3], and [-2, 4, -4].

B

A 3x3 matrix representing a linear transformation.

3x3 real matrices

1. Formula
   Observation

   The expression A o B is written to denote the composition of the transformations represented by matrices A and B.

A o B

The composition of the linear transformations represented by matrices A and B.

3x3 real matrices

1. Formula
   Observation

   The vector [1, 0, 0] is used as the first standard basis vector.

e_1

The first standard basis vector in R^3.

R^3

1. Formula
   Observation

   The vector [0, 1, 0] is used as the second standard basis vector.

e_2

The second standard basis vector in R^3.

R^3

1. Formula
   Observation

   The vector [0, 0, 1] is used as the third standard basis vector.

e_3

The third standard basis vector in R^3.

R^3

1. Audio
   Observation

   The explanation fixes the order: B acts before A.

2. Formula
   Observation

   The displayed notation is A ∘ B.

In this clip, A ∘ B denotes the transformation obtained by applying B first and then applying A to the result. The visual notation uses a small circle between A and B rather than ordinary juxtaposition.

1. A and B are treated as transformations on the same space, here three-dimensional space.

2. The output of B must be an input acceptable to A.

1. Audio
   Observation

   The construction treats a column of B as an input to A and places its image in the corresponding result column.

2. Animation
   Observation

   Purple ovals mark the first column of B and the first column of A∘B; orange ovals mark the third column of B and the third column of A∘B; a red oval marks the middle column of B.

The matrix for A∘B can be built column by column: the j-th column of A∘B is the image under A of the j-th column of B. The clip demonstrates this by pairing the known first and third columns visually before focusing on the missing middle column.

1. A is a linear transformation represented by a matrix.

2. Columns are read as vectors in the domain of A.

1. Composition order for A ∘ B

1. Formula
   Observation

   The middle column vector [0, 2, 3]^T is rewritten as 0[1,0,0]^T + 2[0,1,0]^T + 3[0,0,1]^T.

2. Audio
   Observation

   The input is separated into its coordinate-direction components before applying the transformation.

The vector [0, 2, 3]^T is expressed as a linear combination of the standard basis vectors in R^3. This prepares the next step, where the transformation A is applied to each basis-vector component instead of to the raw coordinates.

1. The vectors are in R^3.

2. The coefficients are the coordinates of the vector relative to the standard basis.

1. Audio
   Observation

   The explanation replaces each coordinate direction by its transformed image while retaining the coefficients.

2. Formula
   Observation

   The board still shows the decomposition into standard basis vectors immediately before this statement.

1. Within this item’s initial interval: The clip ends before the images A e_1, A e_2, A e_3 are explicitly written out or combined into the final middle column. The later part of the complete video performs the substitution and numerical sum.

Linearity allows the same coefficients to be used when combining the images of the basis vectors under A. This part states the rule; the remainder of the video substitutes the columns and performs the arithmetic.

1. A is linear: applying it preserves sums and scalar multiples.

2. The e_i are basis vectors in the domain of A.

1. Writing a vector using standard basis vectors

1. Audio
   Observation

   The continued explanation uses the input matrix columns instead of single coordinate directions.

2. Formula
   Observation

   The expression A o B is shown, and the columns of B are used to compute the resulting matrix.

To find the matrix representation of the composition A o B, apply the transformation A to each column of B. The resulting vectors form the columns of the matrix A o B.

1. A and B are both 3x3 matrices.

2. The operation is performed over the real numbers.

3. These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.

1. A
2. B
3. A o B

1. Audio
   Observation

   The explanation interprets the two displayed matrices and their composition as transformations of the same space.

2. Formula
   Observation

   Three 3×3 arrays are displayed: A, B, and a partially completed A∘B.

Each of the displayed 3×3 matrices may be viewed as a transformation in three-dimensional space, and the composition A∘B can also be represented by a 3×3 matrix.

1. The matrices are 3×3.

2. They are being interpreted as transformations on three-dimensional space.

3. These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.

For the two displayed matrices A and B and their composition A∘B in this example.

1. Audio
   Observation

   The given outer columns illustrate the same column-image rule used for the unfilled column.

2. Animation
   Observation

   The first and third columns are circled in matching colors across B and A∘B.

The first column [6,4,-13]^T of A∘B is the result of applying A to the first column [-2,-3,2]^T of B, and the third column [6,14,2]^T of A∘B is the result of applying A to the third column [-2,4,-4]^T of B.

1. A and B are the displayed matrices.

2. Columns are interpreted as input vectors to the transformation A.

For the first and third columns explicitly shown in the displayed matrices.

1. Audio
   Observation

   The setup connects the missing column to a standard-basis decomposition and transformed basis images.

2. Formula
   Observation

   The board shows A, B, the partial A∘B matrix, and the decomposition 0[1,0,0]^T + 2[0,1,0]^T + 3[0,0,1]^T.

3. Animation
   Observation

   Colored ovals connect columns of B to columns of A∘B and isolate the middle column of B as the target.

1. Within this item’s initial interval: The derivation stops before the final numerical middle column is computed on screen. The later part of the complete video performs the substitution and numerical sum.

2. Within this item’s initial interval: The explicit values of A e_1, A e_2, and A e_3 are not written within this clip. The later part of the complete video performs the substitution and numerical sum.

1. Expression

   $$B_{:,2}=\begin{bmatrix}0\\2\\3\end{bmatrix}$$
   Explanation

   Identify the middle column of B as the vector that must be transformed by A.

   Justification

   Read directly from the displayed matrix B.

   Shown in the video
2. Expression

   $$(A\circ B)_{:,2}=A\left(\begin{bmatrix}0\\2\\3\end{bmatrix}\right)$$
   Explanation

   By the column-wise rule for composition, the missing middle column of A∘B is the image under A of the middle column of B.

   Justification

   Uses the column-image rule and linearity explained in the lesson.

   Shown in the video
3. Expression

   $$\begin{bmatrix}0\\2\\3\end{bmatrix}=0\begin{bmatrix}1\\0\\0\end{bmatrix}+2\begin{bmatrix}0\\1\\0\end{bmatrix}+3\begin{bmatrix}0\\0\\1\end{bmatrix}$$
   Explanation

   Rewrite the input vector as a linear combination of standard basis vectors.

   Justification

   Uses the column-image rule and linearity explained in the lesson.

   Shown in the video
4. Expression

   $$A\left(0e_1+2e_2+3e_3\right)=0A(e_1)+2A(e_2)+3A(e_3)$$
   Explanation

   Replace each basis vector by its image under A and pull out the scalar coefficients.

   Justification

   Uses the column-image rule and linearity explained in the lesson.

   Shown in the video
5. Expression

   $$A(e_1)=\begin{bmatrix}-3\\-3\\3\end{bmatrix},\quad A(e_2)=\begin{bmatrix}0\\-2\\3\end{bmatrix},\quad A(e_3)=\begin{bmatrix}0\\-4\\1\end{bmatrix}$$
   Explanation

   The needed images are the columns of A, so the computation could continue from the displayed matrix A.

   Justification

   Editorial forward calculation from the displayed A; explicitly worked out later in the same full video.

   Supplementary explanation
6. Expression

   $$0\begin{bmatrix}-3\\-3\\3\end{bmatrix}+2\begin{bmatrix}0\\-2\\3\end{bmatrix}+3\begin{bmatrix}0\\-4\\1\end{bmatrix}=\begin{bmatrix}0\\-16\\9\end{bmatrix}$$
   Explanation

   Carrying the stated method to completion gives the missing middle column.

   Justification

   Editorial forward calculation from the displayed A; explicitly worked out later in the same full video.

   Supplementary explanation

This initial interval establishes the column-image setup. Its editorial forward calculation gives [0,-16,9]^T; the later interval explicitly works out and displays the same middle column.

1. Audio
   Observation

   The calculation weights each column of A by its corresponding input coordinate, then adds the vectors.

2. Formula
   Observation

   The calculation is shown as 0*[-3,-3,3] + 2*[0,-2,3] + 3*[0,-4,1], which simplifies to [0, -16, 9].

1. Expression

   $$0 \cdot \begin{bmatrix} -3 \\ -3 \\ 3 \end{bmatrix} + 2 \cdot \begin{bmatrix} 0 \\ -2 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} 0 \\ -4 \\ 1 \end{bmatrix}$$
   Explanation

   Apply the transformation A to the second column of B, which is [0, 2, 3]. This is done by taking the linear combination of the columns of A using the entries of the second column of B as coefficients.

   Justification

   Definition of matrix-vector multiplication.

   Shown in the video
2. Expression

   $$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ -4 \\ 6 \end{bmatrix} + \begin{bmatrix} 0 \\ -12 \\ 3 \end{bmatrix}$$
   Explanation

   Compute the scalar multiplications for each term.

   Justification

   Scalar multiplication of vectors.

   Shown in the video
3. Expression

   $$\begin{bmatrix} 0+0+0 \\ 0-4-12 \\ 0+6+3 \end{bmatrix} = \begin{bmatrix} 0 \\ -16 \\ 9 \end{bmatrix}$$
   Explanation

   Add the resulting vectors component-wise to get the final column.

   Justification

   Vector addition.

   Shown in the video

The second column of the matrix A o B is [0, -16, 9].

1. Formula
   Observation

   The problem is displayed as A=[[-3,0,0],[-3,-2,-4],[3,3,1]], B=[[-2,0,-2],[-3,2,4],[2,3,-4]], and A∘B=[[6,_,6],[4,_,14],[-13,_,2]].

2. Audio
   Observation

   The prompt asks the viewer to determine the blank column of the composition matrix.

1. Within this item’s initial interval: The worked example is only set up in this clip; the final middle-column entry is not written before the end. The later part of the complete video performs the substitution and numerical sum.

Given two 3×3 matrices A and B and a partially completed matrix for A∘B with the first and third columns filled in, determine the missing middle column.

1. $A=\begin{bmatrix}-3&0&0\\-3&-2&-4\\3&3&1\end{bmatrix}$

2. $B=\begin{bmatrix}-2&0&-2\\-3&2&4\\2&3&-4\end{bmatrix}$

3. $A\circ B=\begin{bmatrix}6&-&6\\4&-&14\\-13&-&2\end{bmatrix}$

Compute the middle column of A∘B.

1. Expression

   $$(A\circ B)_{:,2}=A\left(B_{:,2}\right)$$
   Explanation

   Use the column-wise interpretation of composition.

   Justification

   Uses the column-image rule and linearity explained in the lesson.

   Shown in the video
2. Expression

   $$B_{:,2}=\begin{bmatrix}0\\2\\3\end{bmatrix}$$
   Explanation

   Read the middle column of B from the displayed matrix.

   Justification

   Direct visual extraction from B.

   Shown in the video
3. Expression

   $$\begin{bmatrix}0\\2\\3\end{bmatrix}=0\begin{bmatrix}1\\0\\0\end{bmatrix}+2\begin{bmatrix}0\\1\\0\end{bmatrix}+3\begin{bmatrix}0\\0\\1\end{bmatrix}$$
   Explanation

   Decompose the input vector into standard basis vectors.

   Justification

   Written on the board and explained in the audio.

   Shown in the video
4. Expression

   $$A\left(\begin{bmatrix}0\\2\\3\end{bmatrix}\right)=0A\left(\begin{bmatrix}1\\0\\0\end{bmatrix}\right)+2A\left(\begin{bmatrix}0\\1\\0\end{bmatrix}\right)+3A\left(\begin{bmatrix}0\\0\\1\end{bmatrix}\right)$$
   Explanation

   Apply A to each basis-vector component.

   Justification

   Uses the column-image rule and linearity explained in the lesson.

   Shown in the video
5. Expression

   $$=\begin{bmatrix}0\\-16\\9\end{bmatrix}$$
   Explanation

   If the computation is continued using the columns of A, the missing middle column evaluates to this vector.

   Justification

   Editorial forward calculation from the displayed A; explicitly worked out later in the same full video.

   Supplementary explanation

Editorial forward result: the middle column is $\begin{bmatrix}0\\-16\\9\end{bmatrix}$. The same calculation is completed on screen later in the full video.

Substitute the completed middle column into A∘B and check that it equals A times the middle column of B; equivalently, verify the full product AB with the displayed A and B.

1. Formula
   Observation

   The full example shows the computation of A o B by transforming each column of B with A. The first and third columns are pre-filled, and the second column is calculated during the clip.

Given matrices A = [[-3, 0, 0], [-3, -2, -4], [3, 3, 1]] and B = [[-2, 0, -2], [-3, 2, 4], [2, 3, -4]], find the matrix for the composition A o B.

1. Matrix A is provided.

2. Matrix B is provided.

3. The first column of A o B is already computed as [6, 4, -13].

4. The third column of A o B is already computed as [6, 14, 2].

Compute the second column of the matrix A o B.

1. Expression

   $$A \mathbf{b}_2 = 0 \cdot \mathbf{a}_1 + 2 \cdot \mathbf{a}_2 + 3 \cdot \mathbf{a}_3$$
   Explanation

   Identify the second column of B as [0, 2, 3] and express the transformation as a linear combination of the columns of A.

   Justification

   Method for matrix composition via column transformation.

   Shown in the video
2. Expression

   $$0 \cdot \begin{bmatrix} -3 \\ -3 \\ 3 \end{bmatrix} + 2 \cdot \begin{bmatrix} 0 \\ -2 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} 0 \\ -4 \\ 1 \end{bmatrix}$$
   Explanation

   Substitute the actual column vectors from matrix A into the linear combination.

   Justification

   Substitution.

   Shown in the video
3. Expression

   $$\begin{bmatrix} 0 \\ -4 \\ 6 \end{bmatrix} + \begin{bmatrix} 0 \\ -12 \\ 3 \end{bmatrix}$$
   Explanation

   Perform the scalar multiplications. The first term becomes the zero vector and can be omitted.

   Justification

   Scalar multiplication and properties of the zero vector.

   Shown in the video
4. Expression

   $$\begin{bmatrix} 0 \\ -16 \\ 9 \end{bmatrix}$$
   Explanation

   Add the two resulting vectors to get the final second column.

   Justification

   Vector addition.

   Shown in the video

The second column of A o B is [0, -16, 9]. The full matrix is [[6, 0, 6], [4, -16, 14], [-13, 9, 2]].

The result is placed into the pre-drawn matrix structure for A o B, completing the example.

1. Diagram
   Observation

   A black background shows three handwritten matrices: white A at upper left, red B at upper right, and white A∘B at lower left with blank middle-column placeholders.

2. Animation
   Observation

   A yellow pointer dot moves among A, B, and the blank middle column while the speaker introduces the problem.

1. Matrix A

2. Matrix B

3. Partial matrix A∘B

4. Yellow pointer dot

1. The pointer shifts attention from A and B to the blank middle column of A∘B.

1. The entries of A and B remain fixed.

2. The first and third columns of A∘B remain visible while the middle column stays blank.

The visual layout establishes the task: infer the missing middle column of the composed transformation matrix from the given matrices.

1. Animation
   Observation

   A purple oval is drawn around the first column of B and then around the first column of A∘B.

2. Animation
   Observation

   An orange oval is drawn around the third column of B and then around the third column of A∘B.

3. Animation
   Observation

   A red oval is drawn around the middle column of B.

1. First column of B

2. Third column of B

3. Middle column of B

4. First column of A∘B

5. Third column of A∘B

6. Purple oval

7. Orange oval

8. Red oval

1. Matching columns are grouped by color to show correspondence.

2. Attention moves from already-known columns to the unknown middle column.

1. The numerical entries inside the matrices do not change.

2. The order of columns remains first, middle, third.

The animation makes explicit the rule that each column of A∘B is obtained by applying A to the corresponding column of B.

1. Formula
   Observation

   To the right of the matrices, the speaker writes 0[1,0,0]^T + 2[0,1,0]^T + 3[0,0,1]^T.

2. Animation
   Observation

   The yellow pointer hovers over the newly written basis-vector terms as they are added.

1. Within this item’s initial interval: No further written substitution of A(e_i) appears before the clip ends. The later part of the complete video performs the substitution and numerical sum.

1. Vector [0,2,3]^T implicitly referenced

2. Standard basis vectors [1,0,0]^T, [0,1,0]^T, [0,0,1]^T

3. Scalar coefficients 0, 2, 3

1. The single column vector is replaced by an explicit linear combination of basis vectors.

1. The represented vector remains [0,2,3]^T.

2. The matrices A, B, and A∘B stay on the left side of the board.

This visual step prepares the application of linearity: transform each basis direction separately and recombine with the same coefficients.

1. Animation
   Observation

   Colored circles (red, purple, yellow) appear around the columns of matrix A and the corresponding standard basis vectors, then shift to highlight the columns of matrix B.

1. Matrix A

2. Matrix B

3. Standard basis vectors

4. Colored circles

1. Circles move from highlighting the standard basis vectors to highlighting the columns of matrix B.

1. The matrices A and B remain static on the board.

This visual aid demonstrates that the method of finding the matrix for a transformation by applying it to basis vectors is generalized here by applying transformation A to the columns of matrix B.

1. Animation
   Observation

   The speaker writes out the linear combination for the second column of A o B, performs the scalar multiplications, and adds the results, filling in the final values.

1. Handwritten equations

2. Vectors

3. Matrix A o B

1. Equations are written sequentially.

2. Values are calculated and filled into the matrix A o B.

1. The initial setup of matrices A and B remains visible.

This event shows the step-by-step arithmetic process of computing one column of the resulting composite matrix.

1. Audio
   Observation

   The explanation specifies that the right-hand transformation acts first in this notation.

One might think A∘B means apply A first and then B because A is written first.

In this clip, A∘B is defined as applying B first and then A, so the rightmost transformation acts first.

1. Audio
   Observation

   The basis directions must be replaced by their images before recombining the components.

After writing a vector as a combination of standard basis vectors, one might incorrectly leave the basis vectors unchanged and only multiply the coefficients.

The correct next step is to replace each basis vector e_i by A(e_i), then recombine with the same scalar coefficients.

1. Audio
   Observation

   The stated composition order leads to the column-image construction.

2. Animation
   Observation

   Column pairings are highlighted immediately after the composition order is introduced.

The definition of composition as “B first, then A” is applied to obtain the practical rule that each column of A∘B is A applied to the corresponding column of B.

1. Audio
   Observation

   The required input column is next expanded in coordinate directions.

2. Formula
   Observation

   The decomposition appears directly after the discussion of applying A to the middle column.

To compute A applied to a specific column, the clip reduces the problem to transforming standard basis directions and recombining them.

1. Audio
   Observation

   The recombination keeps each coefficient and transforms its associated basis direction.

1. Within this item’s initial interval: The word linearity is not spoken explicitly in this excerpt. The later part of the complete video performs the substitution and numerical sum.

The step from a basis decomposition to transforming each term relies on the linear behavior of A, even though the clip states the procedure rather than naming the property.

1. Audio
   Observation

   The explanation compares a matrix’s basis images with the column inputs needed for the composition.

The method of finding a transformation matrix by applying it to standard basis vectors is generalized to finding the composition A o B by applying A to the columns of B.

1. Audio
   Observation

   The explanation establishes the order of the two transformations.

1. Composition order for A ∘ B

1. Audio
   Observation

   The construction obtains result columns from the images of input columns.

2. Animation
   Observation

   Corresponding columns are circled in matching colors.

1. Columns of A∘B come from applying A to columns of B

1. Formula
   Observation

   The vector [0,2,3]^T is rewritten as 0e_1+2e_2+3e_3.

2. Audio
   Observation

   The explanation uses transformed coordinate directions to carry out the computation.

1. Writing a vector using standard basis vectors
2. Transform the basis directions and recombine

1. Audio
   Observation

   At this point the method has been set up; the subsequent part performs the arithmetic.

2. Formula
   Observation

   The displayed A contains the columns needed to continue the computation.

1. Within this item’s initial interval: This answer is not shown in the clip; it is obtained by continuing the displayed method. The later part of the complete video performs the substitution and numerical sum.

1. Columns of A∘B come from applying A to columns of B
2. Transform the basis directions and recombine

1. Audio
   Observation

   The continued worked example demonstrates the column-image method.

1. Matrix Composition via Column Transformation
2. Computing the Composition of Two 3x3 Matrices

1. Audio
   Observation

   The composition supplies outputs from B as inputs to A.

1. Matrix Composition via Column Transformation
2. s109-cr-basis-to-columns