# Finding $\ker$, ${\rm Im}$, $\dim$ of a linear transformation

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 $${\rm Ker}(\alpha)$$, $${\rm Im}(\alpha)$$, $$\text{dim } {\rm Ker}(\alpha)$$, and $$\text{dim } {\rm Im}(\alpha)$$.

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 $\Bbb R^3$?

• Hope you're holding up despite all the grading! Commented Jun 25, 2014 at 13:02
• Ok, by my definition, the dimension is the number of elements in a basis, so does that mean that the dimension of ker(alpha) is 1 and the dimension of im(alphs) is 2?
– cele
Commented Jun 25, 2014 at 18:41

The kernel of a linear transformation $\alpha$ is the collection of vectors $X$ such that $\alpha(X)$ is the zero vector. In your example, the kernel is the collection of vectors $(a,b,c)\in \mathbb{R}^3$ such that $$\alpha\begin{bmatrix}a\\b\\c \end{bmatrix}=\begin{bmatrix}a+b+c\\-a-c\\b \end{bmatrix} = \begin{bmatrix} 0\\ 0\\0 \end{bmatrix}.$$

Two vectors are equal if their components are equal. So, how should you find the correct components $(a,b,c)$?

When you find the kernel, find a basis for it. How many vectors are in the basis? You can use this theorem to answer another question you've posed.

The answer to this question should help you find the image of $\alpha$. Recall that the image of a linear transformation $\alpha: V\rightarrow V$ is the collection of vectors $v\in V$ such that $\alpha(w) = v$ for some $w\in V$.

You already know what the image of a particular vector $(a,b,c)$ looks like under $\alpha$. Can you find a way to write all such vectors as the linear combination of basis vectors?

Here is a start but after that you need to read through the basic definitions to solve these.

First find the matrix representation of $\alpha$, which is as given below $$A=\begin{bmatrix}1&1&1\\-1&0&-1\\0&1&0\end{bmatrix}.$$

Now you use row reduction and definitions to solve it. For example, kernel means all those vectors which map to the zero vector so it is equivalent to solving the homogeneous system $Ax=0$.

• Someone down voted this response. Just curious about the reason. Commented Feb 20, 2016 at 20:26