Is there any numerical algorithm to find a basis for the subspace, $T^{-1}(\text{im} S)=\{x: T(x)\in \text{im }S\}$ where $T,S:\mathbb{R}^4\to\mathbb{R}^4$ linear map with given matrices $A,E$ as their representation, preferably not invertible. I thought this way: Suppose $v_1,\dots,v_k$ be bas...

Is there any numerical algorithm to find a basis for the subspace, $T^{-1}(\text{im} S)=\{x: T(x)\in \text{im }S\}$ where $T,S:\mathbb{R}^4\to\mathbb{R}^4$ linear map with given matrices $A,E$ as their representation, preferably not invertible. I thought this way: Suppose $v_1,\dots,v_k$ be bas...

I've to find the Basis of $W=\operatorname{span}\{v_1,v_2,v_3,v_4,v_5\} \subseteq\mathbb{F}_{2} ^{5}$ $v_{1}=(1,0,1,0,1)$ , $v_{2}=(0,0,1,0,1)$ , $v_{3}=(0,0,0,1,0)$ , $v_{4}=(1,0,1,1,1)$ , $v_{5}=(1,0,0,1,0)$ $\mathbb{F}_{2}$ is a Field with $2$ Elements so now first i've wrote a linear combi...

Let $\mathbb{K}[x]_{\leq{n}}$ be the k-vector space for all the single variable polynomials $f(x)$ with $deg(f(x)) \leq n$, and $A=\{f(x) ∈ \mathbb{K}[x]_{\leq n}, x^2 + 1 | f(x)\}$ a linear subspace of $\mathbb{K}[x]_{\leq{n}}$. Find a basis for A. I thought of taking a polynomial $a(x) = a_0+...

Prove that if {v1, v2} is a basis for sp(v1, v2), then a) {v1 + v2, v1 - v2} is also a basis. b) {v1 + v2, v1 - v2, 2v1 - 3v2} is not a basis. a) First, prove that sp(v1, v2) = sp(v1 + v2, v1 - v2). Let vj be a vector in sp(v1, v2), and vj = rv1 + sv2 = [(r+s)/2](v1 + v2) + [(r-s)/2](v1 - v...

We have $A \in \mathbb R ^{\mathrm {mxn} }$ and $B \in \mathbb R ^{\mathrm {nxp}}$ which are two matrices. It is said that $\{ b_1, b_2, ..., b_k\}$, where $k \leq q$, is a basis for $\mathrm{Im}(B)$ and that $\mathrm{Ker}(A)\cap \mathrm{Im}(B) = \{ \vec{0}\}$. We have to show that $\{ Ab_1, ...

The basis was recently created. But it's a horrible tag. There are different notions of basis in mathematics, and they are not entirely the same at all. Hamel basis Hilbert basis. Schauder basis. Topological basis. The tag is used as a free for all. And if it continues to exists, it will be u...

We say that an homogeneous symmetric polynomial $f(x_1,\ldots,x_n)$ of degree $d$ is an $n$-exception if the dimension of the $\mathbb{R}$-span of the following set of polynomials of degree $d-1$ $$A_f=\left\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},\sum_{j=1}^{n}x_{...