13. Quiz 1 Review

What people said

Recommends user9464 · 2017-01 · Mathematics Stack Exchange · on “How can I find the rank of a matrix $A$ when the solution sets to $Ax=b$ is given?”

Watch the video from 10:38 [this video] again. First of all he points out that the size of the matrix $A$ is $3\times 3$. Secondly, the assumption of the complete sets of solutions tells you that $$ Null(A)=2. $$ Now the rank-nullity theorem tells you the rank of $A$ is given by $$ Rank(A)=3-Null(A)=1. $$

Recommends user592938 · 2018-10 · Mathematics Stack Exchange · on “Finding the matrix A in Ax=b”

I am watching the linear algebra course of MIT presented by Gilbert Strang and in Lecture 13 he solves the equation $Ax = b$ . This is not the usual "find $x$ ", instead $x$ and $b$ are given and we need to find the matrix $A$ The link is at [this video] at about minute 14:00. We know: $Ax = b$ $b=\begin{bmatrix}2\\4\\2\end{bmatrix}$ $x=\begin{bmatrix}2\\0\\0\end{bmatrix}$ + $c\begin{bmatrix}1\\1\\0\end{bmatrix}$ + $d\begin{bmatrix}0\\0\\1\end{bmatrix}$ $\dim N(A)=2$ $A$ is a $3\times 3$ matrix. But you can see all of that in the video, it looks much clearer on a blackboard. I am having problems understanding what did professor Gilbert do to get the answer $A=\begin{bmatrix}1&&-1&&0\\2&&-2&&0\\1&

Cites aceminer · 2017-01 · Mathematics Stack Exchange · on “How can I find the rank of a matrix $A$ when the solution sets to $Ax=b$ is given?”

Suppose the complete set of solutions to $Ax = b$ where $b = [2, 4, 2]^T$ is given by $$ x = [2, 0, 0]^T + c[1, 1, 0]^T + d[0, 0, 1]^T. $$ This is from a YouToube Video [this video] of Gilbert Strang's lecture on Linear Algebra. He said that the rank of the matrix $A$ was $1$. I understand that $b$ is a column vector so its rank is $1$. How did the students identify the dimension of the null space of $A$ to be $2$? This is something that baffles me. $$A = \begin{bmatrix}1 & -1 & 0 \\ 2 & -2 & 0 \\ 1 & -1 & 0 \end{bmatrix}$$ Firstly, why is it that if $[0, 0, 1]^T$ is in the null space, the third column is a zero vector. Assuming $c = 1$, $d = 1,$ and $A$ is the matrix as proposed, in the first ro

Cites phoenixSP · 2018-09 · Mathematics Stack Exchange · on “Dimension of the rowspace of a matrix based on x and b”

In lecture 13 of Gibert Strang's lecture videos from MIT 18.06 Linear Algebra Spring 2005, he reviews the following problem: What is the dimension of the rowspace of A where $$ Ax=\begin{bmatrix} 2 \\ 4\\ 2 \\ \end{bmatrix} $$ and $$ x=\begin{bmatrix} 2\\ 0\\ 0\\ \end{bmatrix} +c\begin{bmatrix} 1\\ 1\\ 0\\ \end{bmatrix} +d\begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix} $$ According to him, the answer is 1 as the nullspace of A has dimension 2. How did he conclude this? Here is the link to the lecture when he starts describing the problem [this video]

Cites tchappy ha · 2022-01 · Mathematics Stack Exchange · on “If $A,B$ have the same $4$ fundamental subspaces, then what can we say about $A$ and $B$? (Gilbert Strang's lecture on YouTube)”

[this video] Let $A, B$ be $m\times n$ matrices. If $A,B$ have the same null space, the same row space, the same column space and the same left null space, then what can we say about $A$ and $B$ ? My answer was " $A$ and $B$ have the same reduced row echelon form". But if $A,B$ have the same null space, then $A$ and $B$ have the same reduced row echelon form. So, I think that Strang's answer in his head is different from my answer and it must be stronger than my answer.