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Pre-Class Assignment: Least Squares Fit (Regression)

Goals for today's pre-class assignment

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1. Least Squares Fit
2. Linear Regression
3. One-to-one and Inverse transform
4. Inverse of a Matrix
5. Assignment Wrap-up

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1. Least Squares Fit

Review Chapters Chapter 13 pg 225-239 of the Boyd textbook.

In this first part of this course, we try to solve the system of linear equations $Ax=b$ with an $m\times n$ matrix $A$ and a column vector $b$.

There are three possible outcomes: an unique solution, no solution, and infinite many solutions. (Review the material on this part if you are no familiar with when the three types of outcomes happen.)

When $m<n$, we call the matrix $A$ underdeterminated, because we can not have an unique solution for it. When $m>n$, we call the matrix $A$ overdeterminated, becasue we may not have a solution with high probability.

However, if we still need to find a best $x$, even when there is no solution or infinite many solutions we use a technique called least squares fit (LSF). Least squares fit find $x$ such that $\|Ax-b\|$ is the smallest (i.e. we try to minimize the estimation error).

• When there is no solution, we want to find $x$ such that $Ax-b$ is small (here, we want $\|Ax-b\|$ to be small).
• If the null space of $A$ is just $\{0\}$, we can find an unique $x$ to obtain the smallest $\|Ax-b\|$.
  • If there is a unique solution $x^*$ for $Ax=b$, then $x^*$ is the optimal $x$ to obtain the smallest $\|Ax-b\|$, which is 0.
  • Because the null space of $A$ is just $\{0\}$, you can not have infinite many solutions for $Ax=b$.
• If the null space of $A$ is not just $\{0\}$, we know that we can always add a nonzero point $x_0$ in the null space of $A$ to a best $x^*$, and $\|A(x^*+x_0)-b\|=\|Ax^*-b\|$. Therefore, when we have multiple best solutions, we choose to find the $x$ in the rowspace of $A$, and this is unique.

**QUESTION 1:** Let $$A=\begin{bmatrix}1\\2\end{bmatrix},\quad b=\begin{bmatrix}1.5 \\ 2\end{bmatrix}$$ Find the best $x$ such that $\|Ax-b\|$ has the smallest value.

Put your answer to the above question here.

**QUESTION 2:** Compute $(A^\top A)^{-1}A^\top b$.

Put your answer to the above question here.

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2. Linear Regression

Watch the video for using Least Squares to do linear regression.

from IPython.display import YouTubeVideo
YouTubeVideo("Lx6CfgKVIuE",width=640,height=360, cc_load_policy=True)

**QUESTION 3:** How to tell it is a good fit or a bad one?

Put your answer to the above question here.

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3. One-to-one and Inverse transform

Read Section 4.9 of the textbook if you are not familiar with this part.

Definition: A transformation $T:U\mapsto V$ is said to be one-to-one if each element in the range is the image of just one element in the domain. That is, for two elements ($x$ and $y$) in $U$. $T(x)=T(y)$ happens only when $x=y$.

Theorem: Let $T:U\mapsto V$ be a one-to-one linear transformation. If $\{u_1,\dots,u_n\}$ is linearly independent in $U$, then $\{T(u_1),\dots,T(u_n)\}$ is linearly independent in $V$.

Definition: A linear transformation $T:U\mapsto V$ is said to be invertible if there exists a transformation $S:V\mapsto U$, such that $$S(T(u))=u,\qquad T(S(v))=v,$$ for any $v$ in $V$ and any $u$ in $U$.

**QUESTION 4:** If linear transformation $T:U\mapsto V$ is invertible, and the dimension of $U$ is 2, what is the dimension of $V$? Why?

Put your answer to the above question here.

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4. Inverse of a Matrix

• Recall the four fundamental subspaces of a $m\times n$ matrix $A$
  • The rowspace and nullspace of $A$ in $R^n$
  • The columnspace and the nullspace of $A^\top$ in $R^m$

• The two-sided inverse gives us the following $$ {A}{A}^{-1}=I={A}^{-1}{A} $$
  • For this we need $r = m = n$, here $r$ is the rank of the matrix.

• For a left-inverse, we have the following
  • Full column rank, with $r = n \leq m$ (but possibly more rows)
  • The nullspace contains just the zero vector (columns are independent)
  • The rows might not all be independent
  • We thus have either no or only a single solution to $Ax=b$.
  • $A^\top $ will now also have full row rank
  • From $(A^\top A)^{-1}A^\top A = I$ follows the fact that $(A^\top A)^{-1}A^\top$ is a left-sided inverse
  • Note that $(A^\top A)^{-1}A^\top$ is a $n\times m$ matrix and $A$ is of size $m\times n$, theire mulitiplication $(A^\top A)^{-1}A^\top A$ results in a $n\times n$ identity matrix
  • The $A(A^\top A)^{-1}A^\top$ is a $m\times m$ matrix. BUT $A(A^\top A)^{-1}A^\top\neq I$ if $m\neq n$. The matrix $A(A^\top A)^{-1}A^\top$ is the projection matrix onto the column space of $A$.

**QUESTION 5:** What is the projection matrix that projects any vector onto the subspace spanned by $[1,2]^\top$. (What matrix will give the same result as projecting any point onto the vector $[1,2]^\top$.)

Put your answer to the above question here.

**QUESTION 6:** If $m=n$, is the left inverse the same as the inverse?

Put your answer to the above question here.

Theorem: For a matrix $A$ with $r=n<m$, the columnspace of $A$ has dimension $r(=n)$. The linear transfrom $A: R^n\mapsto R^m$ is one-to-one. In addition, the linear transformation $A$ from $R^n$ to the columnspace of $A$ is one-to-one and onto (it means that for any element in the columnspace of $A$, we can find $x$ in $R^n$ such that it equals $Ax$.) Then the left inverse of $A$ is a one-to-one mapping from the columnspace of $A$ to $R^n$, and it can be considered as an inverse transform of $A$.

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3. Assignment wrap-up

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✅ **QUESTION:** How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?

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✅ **QUESTION:** What was the most challenging part of this assignment for you?

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✅ **QUESTION:** What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?

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✅ **QUESTION:** Do you have any further questions or comments about this material, or anything else that's going on in class?

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✅ **QUESTION:** Approximately how long did this pre-class assignment take?

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Congratulations, we're done!

In order to successfully complete this assignment, you must do the required reading, watch the provided videos, and complete all instructions. The embedded survey form must be entirely filled out and submitted on or before 11:59pm on the day before class. Students must come to class the next day prepared to discuss the material covered in this assignment.

Course Resources:

• MTH314 Course Website

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Written by Dr. Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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