Recent content by Zero2Infinity

Homework Statement Consider two vector spaces ##A=span\{(1,1,0),(0,2,0)\}## and ##B=\{(x,y,z)\in\mathbb{R}^3 s.t. x-y=0\}##. Find a basis of ##A\cap B##. I get the solution but I also inferred it without all the calculations. Is my reasoning correct Homework Equations linear dependence...

Homework Statement Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##. Find a matrix ##A## associated to a linear map ##f:\mathbb{R}^3\rightarrow\mathbb{R}^3## through the standard basis such that its nullspace is ##V##...

Thank you again Sir! Just to be sure: ##(1,-4,0),(0,0,1)## is a orthogonal basis of the ##Im(f)## subspace, ##(4,1,0)## is the basis for the one-dimensional nullspace of ##f## and so, by virtue of the rank-nullity theorem, I have an orthogonal basis of the domain of ##f##, i.e. ##\mathbb{R}^3##.

First of all, thank you for your patience. If I add a vector linearly dependent to the other two I get a matrix which kernel is ##(4,1,0)##. For example \begin{equation} \begin{pmatrix}1&-4&0\\0&0&1\\0&0&2\end{pmatrix} \end{equation} Then this is a solution for the excercise? As you might...

Let's see: knowing the basis vectors I can write \begin{equation} \begin{pmatrix}4&a&b\\1&c&d\\0&e&f\end{pmatrix} \begin{pmatrix}1\\-4\\0\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix} \end{equation} Thus I get \begin{equation} \begin{cases}4-4a+0b=0\\1-4c+0d=0\\0-4e+0f=0\end{cases}...

Sorry but I'm not convinced. The basis of the kernel is found by solving the system I wrote, corresponding to the matrix \begin{equation} \begin{pmatrix} 1&-4&0\\0&0&1 \end{pmatrix} \end{equation} This system ha ##\infty^1## solutions of the type ##(4y,y,0)##, i.e. a basis is ##(4,1,0)## and the...

Thank you Sir! :smile:

Homework Statement Build the matrix A associated with a linear transformation ƒ:ℝ3→ℝ3 that has the line x-4y=z=0 as its kernel. Homework Equations I don't see any relevant equation to be specified here . The Attempt at a Solution First of all, I tried to find a basis for the null space by...

Hello everyone! I'm a freshman and currently grinding to prepear the linear algebra test. Hope I can find some help here :)