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, ...
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_{...
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