M A T H 2 1 B
Mathematics Math21b Spring 2009
Linear Algebra and Differential Equations
Exhibit: Duality in Linear Algebra
Office: SciCtr 434

## Row and column picture

One of the most important insights of linear algebra is obtained from the role of columns and rows. The columns are helpful to describe the image, the rows helpful to describe the kernel.

The column picture tells that A x = x1 v1 + ... + xn vn is a linear combination of columns. The row picture tells that A x = 0 means that x is perpendicular to the rows of A. One can state this as
``` ker(A)T = im (AT)
```
which relates the transposed matrix AT of the matrix A with the orthogonal complement VT of the column space V.

## Data

Duality reflects also in statistics If you have two data sets like x = (1,3,1,-2,3) and y = (2,1,-5,5,1). Then there are two ways to describe these data. The first is to see the data as two vectors in 5 dimensional space. The cosine of the angle between X = x-E[x] and Y = y-E[y] is called the correlation of the data. The second dual picture is to visualize the data points (1,2),(3,1),(1,-5),(-2,5),(3,1) in the plane. The regression line passes through the point E[x],E[y] and has the slope m = cos(alpha) ||Y||/||X|| = (X.Y)/||X||2 because Y-m X is minimized if m X is the projection of Y onto X Again, if the data are written as a matrix, then the two pictures correspond to taking the transpose of the matrix.

 ```A = | 1 2 | | 3 1 | | 1 -5 | | -2 5 | | 3 1 | ``` ```AT = | 1 3 1 -2 3 | | 2 1 -5 5 1 | ```
The text passage provides an explanation of duality in linear algebra, focusing on the relationship between the row and column pictures of matrices, and how this concept applies to data sets and statistics. It includes examples and matrix representations to illustrate these points.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is purely expository, explaining concepts and providing examples.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
M A T H 2 1 B
Mathematics Math21b Spring 2009
Linear Algebra and Differential Equations
Exhibit: Duality in Linear Algebra
Office: SciCtr 434

## Row and column picture

One of the most important insights of linear algebra is obtained from the role of columns and rows. The columns are helpful to describe the image, the rows helpful to describe the kernel.

The column picture tells that A x = x1 v1 + ... + xn vn is a linear combination of columns. The row picture tells that A x = 0 means that x is perpendicular to the rows of A. One can state this as
``` ker(A)T = im (AT)
```
which relates the transposed matrix AT of the matrix A with the orthogonal complement VT of the column space V.

## Data

Duality reflects also in statistics If you have two data sets like x = (1,3,1,-2,3) and y = (2,1,-5,5,1). Then there are two ways to describe these data. The first is to see the data as two vectors in 5 dimensional space. The cosine of the angle between X = x-E[x] and Y = y-E[y] is called the correlation of the data. The second dual picture is to visualize the data points (1,2),(3,1),(1,-5),(-2,5),(3,1) in the plane. The regression line passes through the point E[x],E[y] and has the slope m = cos(alpha) ||Y||/||X|| = (X.Y)/||X||2 because Y-m X is minimized if m X is the projection of Y onto X Again, if the data are written as a matrix, then the two pictures correspond to taking the transpose of the matrix.

 ```A = | 1 2 | | 3 1 | | 1 -5 | | -2 5 | | 3 1 | ``` ```AT = | 1 3 1 -2 3 | | 2 1 -5 5 1 | ```
user
The text passage provides an explanation of duality in linear algebra, focusing on the relationship between the row and column pictures of matrices, and how this concept applies to data sets and statistics. It includes examples and matrix representations to illustrate these points.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is purely expository, explaining concepts and providing examples.