What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional subspace of $V$, and denote the projection from $V$ to $W$ by $P_W$. Let $V_1\supseteq V_2\supseteq V_3\supseteq\cdots$ be a decreasing sequence of […]

Let $v_1, \dots, v_m$ be a linearly dependent list of vectors. If $v_1 \ne 0$, then there is some $v_j$ in the span of $v_1, \dots, v_{j-1}$ If we let j be the smallest integer with this property, and apply the gram-schmidt procedure to produce an orthonormal list $(e_1, \dots, e_{j-1})$ then $v_j$ is in […]

Let $V, W, U, X$ be $R$-modules where R is a ring. At what level of generality, if any is it true that the maps (I always mean linear) from $V \otimes W$ to $U \otimes X$ can be identified with $L(V, U)\otimes L(W, X)$ where $L(., .)$ is the space of maps, via the […]

If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e. $L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$, is there an easy way to determine a basis for $L_1\cap L_2$? Answers of the form “Plug the matrices into a computer and ask for Hermite Normal Form, etc” are perfectly acceptable as this is […]

I want to prove that if $W$ is a subspace of an inner product space $V$, then $\dim W + \dim W^\perp = \dim V$. I have defined $x^\perp = \{ y : x \cdot y = 0\}$, where $\cdot$ denotes the dot product. It is a pretty elementary result, but I’m not sure how […]

Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and suppose $T$ is a nonsingular linear transformation of $V$ such that $T^{-1} = T^2 + T$. Prove that the dimension of $V$ is divisible by $3$. If the dimension of $V$ is precisely $3$, prove that all such transformations $T$ are similar. So applying […]

An exercise from the book I am reading is: “Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus.” I have tried many systems. I found one quickly but I was lucky even if I had a few clues thanks my previous trials. I wonder if there […]

[Ciarlet 1.2-2] Let $O$ be an orthogonal matrix. Show that there exists an orthogonal matrix $Q$ such that $$Q^{-1}OQ\ =\ \left(\begin{array}{rrrrrrrrrrr} 1 & & & & & & & & & & \\ & \ddots & & & & & & & & & \\ & & 1 & & & & & & & […]

Suppose $A$ is a $m\times n$ matrix, $x$ is a vector of unkowns and $b$ is a vector of know entries. Why don’t $$Ax=b$$ and $$A^TAx = A^Tb$$ have the same solution ($x$)? It seems to me that I could get from the first equation to the second equation by simply multiplying both sides by […]

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