Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Determine a basis of $\operatorname{Ker} F$ and one for $\operatorname{Im} F$, where $F:\Bbb R^4\to \Bbb R^3$ is the linear transformation defined by $$F(x_1,x_2,x_3,x_4):=(x_1+x_2+x_3+x_4, 2x_2+x_3+x_4,4x_2+2x_3+2x_4) .$$

I have no idea how to start. Any idea please? Thank you.

share|cite|improve this question
up vote 2 down vote accepted

You could first find the matrix representation $A$ of $F$.

$A$ is the $3\times 4$ matrix whose $i^{\rm th}$ column is $F({\bf e}_i)$where ${\bf e}_i $ is the $i^{\rm th}$ unit vector in $\Bbb R^4$. You then have $$ F({\bf x})=A{\bf x}, $$ for all ${\bf x}\in \Bbb R^4$.

Then a basis for $\text{ ker}( F)$ is given by a basis for the null space of $A$ and a basis for the image of $F$ is given by a basis of the column space of $A$.

share|cite|improve this answer

Let's start with the kernel. The kernel is the set of inputs yielding the output zero. So for us that's the set of solutions to the system, $$\eqalign{x_1+x_2+x_3+x_4&=0\cr2x_2+x_3+x_4&=0\cr4x_2+2x_3+2x_4&=0\cr}$$ Do you know how to find the solutions of such a homogeneous system of linear equations? Do you know how to find a basis for that set of solutions? If not, better learn it fast, as you'll be using that technique over and over and over in linear algebra.

By the way, there's nothing wrong with posting homework problems to this website, but, if this is a homework problem, you ought to add the "homework" tag.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.