1
$\begingroup$

I have this problem:

enter image description here

I've attempted calculating: $[L(v_1)]_T$ but cannot understand where I'm going wrong. Apparently the answer is (1, -1). Here is my attempt:

enter image description here

Can anyone spot where I've gone wrong? I think I did something wrong in the transformation but I'm not sure why.

$\endgroup$
  • $\begingroup$ You can typeset mathematics with MathJax. $\endgroup$ – Ennar Nov 6 '18 at 21:56
  • $\begingroup$ $\;Lv_i\;$ is simply the $\;i\,-$ th column in the matrix representing $\;L\;$ wrt those basis: this is precisely the definition of matrix wrt basis! $\endgroup$ – DonAntonio Nov 6 '18 at 21:58
  • $\begingroup$ @DonAntonio, true. But I'd say this is a clever exercise since knowing definition and understanding it are different things. Even more so if you don't really know definition precisely. $\endgroup$ – Ennar Nov 6 '18 at 22:12
  • $\begingroup$ @Ennar And that is why I remarked that that is precisely how we define this thing. A careful student will go and read his notes. $\endgroup$ – DonAntonio Nov 6 '18 at 23:05
3
$\begingroup$

Hint: If $A$ is matrix of $L$ in pair of bases $S$ and $T$, to calculate $Lv_1$ using $A$ you first need to write $v_1$ in basis $S$.

$\endgroup$
  • $\begingroup$ That's the thing that confuses me. Isn't $v_1$ a vector that is part of the basis S? Is it somehow saying to write the vector in terms of the basis it makes up? $\endgroup$ – dj1121 Nov 7 '18 at 0:39
  • $\begingroup$ @dj1121, let me ask you, what do you think $[-1, 1, 0]$ stands for? $\endgroup$ – Ennar Nov 7 '18 at 0:47
  • $\begingroup$ Ahh wait a minute. Ok so I know that the columns of the matrix representing the transformation are just the transformation applied to the column vectors in the basis. In this case they're coming from S but how should I know that they come from S? Why not from T? Is that because the basis S is a basis for R3 and the transformation goes from R3 -> R2? So that means [-1,1,0] is $v_1$ before you apply the transformation so it's $[v_1]_s$ right? $\endgroup$ – dj1121 Nov 7 '18 at 0:56
  • $\begingroup$ @dj1121, let's step back a bit. $[-1,1,0]$ is just a convenient way to express a vector in a basis. Since it doesn't say otherwise, you should assume it's canonical basis $\{e_1,e_2,e_3\}$. Thus, $[-1,1,0]$ represents $-e_1 + e_2.$ But, your matrix accepts vector columns in basis $S$, not in canonical basis, so you multiplying $A\, [-1,1,0]$ calculates $L(-v_1+v_2)$, not $Lv_1$. What you need to do is find a column vector $[a,b,c]$ such that $av_1+bv_2+cv_3 = v_1$ and then calculate $A\, [a,b,c]$ to get $Lv_1$. $\endgroup$ – Ennar Nov 7 '18 at 1:01
  • $\begingroup$ What I want to say is that you know that $v_1$ is represented by $[-1,1,0]$ in canonical basis. But when you give that column vector to matrix $A$, it doesn't know that you mean vector $v_1$, it thinks that $[-1,1,0]$ is in basis $S$, so for $A$ column $[-1,1,0]$ means $-v_1+v_2$. You have to write $v_1$ in a way that $A$ understands. $\endgroup$ – Ennar Nov 7 '18 at 1:10
0
$\begingroup$

HINT

In the given basis

$$L(v_1)=A\cdot v_1=[1,-1]=1\cdot w_1-1\cdot w_2$$

and so on.

$\endgroup$

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.