I have a set of three matrices and want to know whether or not they form a basis for $ℝ^{3\times3}$

Any help is appreciated

the matrices: \begin{bmatrix}1&0&0\\0&1&0\\0 &0&1\end{bmatrix}\begin{bmatrix}0&1&0\\0&0&1\\0 &0&0\end{bmatrix}\begin{bmatrix}0&0&1\\0&0&0\\0 &0&0\end{bmatrix}

I know I have to show they are linearly independent and span the vector space, but I really don't get how I should do that

Thanks in advance

  • 6
    $\begingroup$ What is the dimension of $\mathbb{R}^{3\times 3}$? $\endgroup$ – Ofir Jan 9 '17 at 11:05
  • $\begingroup$ I think 3, right? $\endgroup$ – Amaluena Jan 9 '17 at 11:06
  • $\begingroup$ You have 9 entries in each matrix, so you should expect that the dimension will be 9, and in particular 3 matrices cannot span the entire space. $\endgroup$ – Ofir Jan 9 '17 at 11:07
  • $\begingroup$ @Amaluena I guess you were thinking of each column or row as a vector, no? $\endgroup$ – Billy Rubina Jan 9 '17 at 11:50
  • $\begingroup$ i may be wrong but if see more precisely 1st vector of your basis spans a three dimensional vector space(say V) and 2nd vector spans 2 dimensional vector space possibly contained in V and 3rd spans 1 dimensional vector space also contained in V if one can prove 1st basis vector is l.c of other two $\endgroup$ – Kislay Tripathi Jan 11 '17 at 12:02

Denote $C_1,C_2$ and $C_3$ the $3 \times 3$ matrices from your question. Is it true that $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \in \text{span}(C_1,C_2,C_3)?$$

  • $\begingroup$ No, they don't, I see now they never span the entire space $\endgroup$ – Amaluena Jan 9 '17 at 11:12

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.