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Answer by Kat for Find the matrix of a linear transformation given kernel
Notice that $ker(F)=\left\langle (1, 0, -1), (0,1,2) \right\rangle$ (why?) and also notice that (1,0-1) and (0,1,2) are linearly independent therefore using the Dimension Theorem :$$3=dim(\mathbb...
View ArticleAnswer by DonAntonio for Find the matrix of a linear transformation given kernel
I will take the most basic, elementary way I can think of. First, we work all the time with the standard basis in both domain and codomain.Second, since $\;F(0,1,1)=(-1,-3,0,1)\;$, this means both...
View ArticleAnswer by the_candyman for Find the matrix of a linear transformation given...
Let's call $M$ one matrix associated to $F$.We know that $F(v) = 0$ for any $v \in \ker(F)$. This means that each row of $M$ is orthogonal to any vector $v \in \ker(F)$.A basis of $\ker(F)$ is:$$k_1 =...
View ArticleAnswer by InsideOut for Find the matrix of a linear transformation given kernel
You can start choosing a basis for $ker F$, for instance $\{(1,0,-1),(2,1,0)\}$ and complete it to a basis of $\Bbb R^3$ adding $(0,1,1)$; so we have $\mathcal{B}_{\Bbb...
View ArticleFind the matrix of a linear transformation given kernel
Find the matrix associated to a linear function:$ F: \Bbb R^3 \to \Bbb R^4 $$ \ker F=\left\{(x,y,z) \in \Bbb R^3 \mid x - 2y + z = 0\right\} $and $F((0,1,1))=(-1,-3,0,1)$It's a multiple choice...
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