## Lecture 1: The Column Space of A Contains All Vectors Ax

## 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 **R**^{4} 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 **R**^{3}. 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 \(\left[\begin{array}{cc}0&A\\0&A\\\end{array}\right]\) ？

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