Intereting Posts

Do Cantor's Theorem and the Schroder-Bernstein Theorem Contradict?
How $\frac{1}{n}\sum_{i=1}^n X_i^2 – \bar X^2 = \frac{\sum_{i=1}^n (X_i – \bar X)^2}{n}$
Help with the integral for the variance of the sample median of Laplace r.v.
Finding all the numbers that fit $x! + y! = z!$
Game about placing pennies on table
Is $e^{|x|}$ differentiable?
Simplified l'Hospital: please can you check my proof
The series expansion of $\frac{1}{\sqrt{e^{x}-1}}$ at $x=0$
Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$
Finding a specific basis for an endomorphism
Positive operator is bounded
How can we composite two piecewise functions?
$\sin(2\pi nx)$ does not converge for $x \in (0,1/2)$
How to prove there are exactly eight convex deltahedra?
the discriminant of the cyclotomic $\Phi_p(x)$

Suppose $A: n \times n$ and $B: n \times m$ and that $A$ is

invertible.Prove that $\forall \vec{b} \in \mathbb{R}^n, B \vec{x} = \vec{b}$ is

consistent is equivalent to $\forall \vec{b} \in \mathbb{R}^n, (AB)

\vec{x} = \vec{b}$ is consistent.

I’m assuming this boils down to proving that $B$ and $AB$ are equivalent expressions, but I can’t see why this would be the case. Why does $A$ being invertible make the two statements equivalent (that is, why is $A$ invertible required)?

- Find the third vertex of a triangle in $3D$
- Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}$
- Closedness of sets under linear transformation
- Determinant of a adjoint
- For $W \leqslant V$, prove $\dim W + \dim W^\perp = \dim V$
- Find the optimal solution without going through the ERO's

- Let $W$ be a subspace of a vector space $V$ . Show that the following are equivalent.
- Prove: Every Subspace is the Kernel of a Linear Map between Vector Spaces
- Why does this “miracle method” for matrix inversion work?
- Prove that $\operatorname{Trace}(A^2) \le 0$
- Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$
- Fredholm Alternative as seen in PDEs, part 1
- Why is minimizing least squares equivalent to finding the projection matrix $\hat{x}=A^Tb(A^TA)^{-1}$?
- Find the number of $n$ by $n$ matrices conjugate to a diagonal matrix
- If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric
- The annihilator of an intersection is the sum of annihilators

Prove that $b\in$ column space of $B$ if and only if $A^{-1}b\in$ column space of $B$, which is pretty obvious. Because if $b_1,\cdots,b_m$ are the columns of $B$, then $b$ can written as a linear combination, $c_1b_1+\cdots+c_mb_m$ implies $A^{-1}B$ can be written as $c_1A^{-1}b_1+\cdots+c_mA^{-1}b_m$ and vice versa. Note that as $A$ is invertible $A^{-1}b_1,\cdots,A^{-1}b_m$ is a basis for the column space of $B$.

- Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.
- Which is larger? $20!$ or $2^{40}$?
- Darboux's Integral vs. the “High School” Integral
- What is wrong with this funny proof that 2 = 4 using infinite exponentiation?
- Why is Euclid's proof on the infinitude of primes considered a proof?
- Every Banach space is quotient of $\ell_1(I)$
- Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis
- complex integration over the whole plane
- Prove Corollary of the Fundamental lemma of calculus of variations
- Induction proof: n lines in a plane
- Proving a subset is not a submanifold
- Proof : If $f$ continuous in $$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$
- When linear combinations of independent random variables are still independent?
- $f \in {\mathscr R} \implies f $ has infinitely many points of continuity.
- Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition.