[Math] Finding ker, im, dim of a linear transformation

linear algebralinear-transformationsmatrices

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems.

Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the linear transformation given by
$$\alpha\begin{bmatrix}a\\b\\c \end{bmatrix}=\begin{bmatrix}a+b+c\\-a-c\\b \end{bmatrix}$$
Find $Ker(\alpha)$, $Im(\alpha)$, $\text{dim } Ker(\alpha)$, and $\text{dim } Im(\alpha)$.

Best Answer

Let $x=(a,b,c)^T$.

  • We have $$x\in\ker\alpha\iff\alpha(x)=0\iff (b=0)\land(a=-c)\iff x=a(1,0,-1)^T$$ so $$\ker\alpha=\operatorname{span}\left((1,0,-1)^T\right)$$

We have $$\alpha(x)=(a+c)\underbrace{(1,-1,0)^T}_{=u}+b\underbrace{(1,0,1)^T}_{=v}$$ and the two vectors $u$ and $v$ are linearly independent so $$\operatorname{im}(\alpha)=\operatorname{span}(u,v)$$ Can you say what are the dimensions of thees two subspaces of $\Bbb R^3$?

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