18.065 | Spring 2018 | Undergraduate

# Matrix Methods in Data Analysis, Signal Processing, and Machine Learning

Video Lectures

## Description

In this first lecture, Professor Strang introduces the linear algebra principles critical for understanding the content of the course.  In particular, matrix-vector multiplication $$Ax$$ and the column space of a matrix and the rank.

## Summary

Independent columns = basis for the column space
Rank = number of independent columns
$$A = CR$$ leads to: Row rank equals column rank

Related section in textbook: I.1

Instructor: Prof. Gilbert Strang

Problems for Lecture 1

From textbook Section I.1

1. Give an example where a combination of three nonzero vectors in R4 is the zero vector. Then write your example in the form $$A\boldsymbol{x}$$ = 0. What are the shapes of $$A$$ and $$\boldsymbol{x}$$ and 0 ?

4. Suppose $$A$$ is the 3 by 3 matrix ones(3, 3) of all ones. Find two independent vectors $$\boldsymbol{x}$$ and $$\boldsymbol{y}$$ that solve $$A\boldsymbol{x}$$ = 0 and $$A\boldsymbol{y}$$ = 0. Write that first equation $$A\boldsymbol{x}$$ = 0 (with numbers) as a combination of the columns of $$A$$. Why don’t I ask for a third independent vector with $$A\boldsymbol{z}$$ = 0 ?

9. Suppose the column space of an $$m$$ by $$n$$ matrix is all of R3. What can you say about $$m$$? What can you say about $$n$$ ? What can you say about the rank $$r$$ ?

18. If $$A = CR$$, what are the $$CR$$ factors of the matrix $$\left[\begin{array}{cc}0&A\\0&A\\\end{array}\right]$$ ？

## Course Info

Spring 2018
##### Learning Resource Types
Lecture Videos
Problem Sets
Instructor Insights