1
$\begingroup$

Find the basis and dimension of vector space over $\mathbb R$:

a) vector space generated by $\{u+v+w,v+w+z,w+z+u,z+u+v\}$, where $u,v,w,z$ are linearly independent vectors of some vector space

b) $R(A) \cap S(A) \cap Ker(A)$ for matrix $$ A= \begin{pmatrix} 1 & 2 \\ 3 & 5 \\ \end{pmatrix}$$

$R(A)$ is row matrix space, $S(A)$ is column matrix space and $Ker(A)$ is kernel of the matrix.

How to reduce this vector in a) to get the basis? I'm not sure how to calculate $R(A) \cap S(A) \cap Ker(A)$. Will $R(A)$ and $S(A)$ be the same for matrix 2x2? Could someone help me please?

$\endgroup$
2
$\begingroup$

For a), start by checking whether the generators of the vector space are linearly independent. You know that $(u,v,w,z)$ are linearly independent so if $au + bv + cw + dz = 0$ you must have $a = b = c = d = 0$. Write a linear combination of the generators, collect terms and check what you get.

For b), note that $A$ has rank $2$ so $R(A) = S(A) = \mathbb{R}^2$ so you only need to find $\ker(A)$ which amounts to solving a linear system of homogeneous equations. Since $\dim \ker(A) = 2 - \mathrm{rank} A = 0$, you actually don't need to solve a thing.

$\endgroup$
  • 2
    $\begingroup$ Once you know that $A$ has rank $2$, you immediately know that its kernel is trivial. $\endgroup$ – amd Dec 2 '15 at 11:30
  • $\begingroup$ Thanks, I've somehow missed this! $\endgroup$ – levap Dec 2 '15 at 12:15
1
$\begingroup$

Hints:

For (a), What are the coordinates of these vectors in the $\{u,v,w,z\}$ basis? E.g., $u+v+w=(1\;1\;1\;0)^T$. Using these representations, can you think of an easy way to find their span? Think about the row or column space of a matrix.

For (b), you can see by inspection that the rows and columns are linearly independent, so the matrix has full rank. What does this say about the spaces you’re asked to find?

$\endgroup$

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.