0
$\begingroup$

We have that $T:\mathbb R^3 \to \mathbb R^4$ and $T(x_1,x_2,x_3)=(3x_1+4x_2+2x_3,x_1+2x_2,2x_1+x_2+3x_3,-x_1+5x_2-7x_3)$.

I know that if $A$ is any $m \times n$ matrix over a field $\mathbb F$ viewed as a linear map $A:\mathbb F^n \to \mathbb F^m$, then $\operatorname{ker}A=\operatorname{nullsp}A$ and $\operatorname{Im}A=\operatorname{colsp}A$. Then, in this problem, $\ker A$ is described as:

$3x_1+4x_2+3x_3=0$

$x_1+2x_2=0$

$2x_1+x_2+2x_3=0$

$-x_1+5x_2-7x_3=0$

I also know that the dimension of the solution space of a homogeneous system $AX=0$ of linear equations is $s=n-r$, where $n$ is the number of unknowns and $r$ is the rank of the matrix. How would I be able to solve this problem? Thank you.

$\endgroup$
6
  • $\begingroup$ You have to use Gauß's pivot method. You see the system has rank $3$. $\endgroup$
    – Bernard
    Feb 9, 2015 at 0:46
  • $\begingroup$ So, use Gaussian elimination on the kernel system? What result will this give me? $\endgroup$ Feb 9, 2015 at 1:14
  • $\begingroup$ You'll get the dimension of the kernel, and a basis. $\endgroup$
    – Bernard
    Feb 9, 2015 at 1:58
  • $\begingroup$ How does this help me find the desired system of linear equations? $\endgroup$ Feb 9, 2015 at 2:03
  • $\begingroup$ I'm afraid I don't quite get you. Do you want to solve the system of linear equations? $\endgroup$
    – Bernard
    Feb 9, 2015 at 2:15

1 Answer 1

0
$\begingroup$

Consider the images of the canonical basis of $\mathbf R^3$, written as a $3\times 4$ matrix: $$A=\begin{bmatrix}3&4&3\\1&2&0\\2&1&2\\-1&5&-7 \end{bmatrix}$$ You must under which condition on $x,y,z,t$ the system of linear equations: $$A\begin{bmatrix}\lambda\\\mu\\\nu\end{bmatrix}=\begin{bmatrix}x\\y\\z\\t\end{bmatrix}$$ has a solution. Note there are $3$ unknowns and $4$ equations.

Running Gauß's elimination will give you the condition(s) of compatibilty for this system: these conditions are the equations of the image.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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