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 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 \(\left[\begin{array}{cc}0&A\\0&A\\\end{array}\right]\) ？
