8. Solving Ax = b: Row Reduced Form R

What people said

Recommends Gamma Function · 2014-06 · Mathematics Stack Exchange · on “Solve a system of linear equations”

This is the problem of finding the complete solution to $Ax=b$. To learn how to do this, you may want to watch this video [LINK] or Gilbert Strang's full video lecture [this video] or notes [LINK] on the topic. The idea is that we need to find (a) the null space $N$ of $ \left[\begin{array}{rrr|r} 1 & -1 & -2 \\ 2 & 3 & -1 \end{array}\right]$ and (b) a single solution $c$ to $ \left[\begin{array}{rrr} 1 & -1 & -2 \\ 2 & 3 & -1 \end{array}\right]c = \left[\begin{array}{r} 1 \\-2 \end{array}\right] $. Then all the solutions are exactly $x = N + c$. Row reduction is the first step regardless of what we do next. (a) We have $\left[\begin{array}{rrr} 1 & -1 & -2 \\ 0 & 5 & 3 \end{array}\right] \left[\

Cites Tony Tarng · 2016-05 · Mathematics Stack Exchange · on “When does $A^TAX = A^TB$?”

@YvesDaoust I have just learnt the concepts not too long ago on an MIT OCW course 18.06 on linear algebra, and the professor states that in order for the equation $Ax = b$ to always have a solution for every b, $A$ must be of full row rank. Perhaps it is because he is limiting the right hand side of the equation to only contain vectors as oppose to matrix? But $A^Tb$ (NOT B) is a vector as well. This is the link to the Youtube video for reference (around 36:10): [this video]

Cites Bhuwan Bhatt · 2019-07 · Mathematics Stack Exchange · on “How does the rank of matrix affects solvability of $Ax = b$?”

I was going through lecture 8 of Linear algebra by MIT [this video] (at 36:48)i didn't understand the concept where when rank of a matrix is equal to the number of rows ( $r = m$ , for $m\times n$ matrix), the solution for $Ax = b$ exists for every b.Can someone please elaborate on that. Thanks