How do I solve for a matrix using base vectors and vector spaces?

[Kruger]

Homework Statement

The problem is quite easy, but I've still trouble solving this.

Given the two base vectors e1=(1,-2,0) and e2=(0,3,0) and the other ones of a different vector space w1=(1,0,0) and w2=(0,1,0).

I've to find a matrix A that that does the following Ae1=w1 and Ae2=w2

2. The attempt at a solution

Easy isn't it? I've done what the professor did to solve such problems:

Calculate: e1=1*w1+(-2)*w2
and: e2=0*w1+3*w2

thus that shouls yield the matrix A:(1, 0, 0; -2, 3, 0; 0, 0, 0)
where ; is written for different lines in the matrix A.

But if I calculate A*e1 I get something totally wrong.

Where's the mistake in my calculation?

 

[marlon]

You calculated the inverse of A, expressing the e vectors as a linear combination of the w vectors:

Ae1=w1 so e1=A^(-1)w1
Ae2=w2 so e2=A^(-1)w2

Clearly, your answer for A is incorrect since the inverse of your A does NOT exist. For this reason, you need to drop the third dimension. So , for example, e1 becomes (1,-2) etc.

The way you proceed is correct though, except that your A is actually the inverse of A, and the inverse of A is indeed (1,0;-2,3). So acquire A now.

marlon

 

[Kruger]

I don't understand exactly why I've to build the inverse of A, because I search A such that A*e1=w1 not that A*w1=e1.

 

[Kruger]

ahh, k, I got it know, I took your hint marlon, thanks, thanks.