Thread: Consider this matrix and prove...

1. Consider this matrix and prove...

Consider the n x n matrix with all zeros except for the real numbers α1 ... αn on the diagonal. Prove, by checking that it preserve addition and scalar multiplication, that this matrix corresponds to a linear transformation of R^n

That a1 and an are alpha-1 and alpha-n.

I really hope this question makes sense to someone. I'm not the strongest at math and I need this course to get into a Master's program. I really hope you can help me!

Thanks so much!
- H.

Edit: I posted this in another form area by accident, and I hope it's ok if I reposted it here. :/

2. Re: Consider this matrix and prove...

Originally Posted by hyrule
Consider the n x n matrix with all zeros except for the real numbers α1 ... αn on the diagonal. Prove, by checking that it preserve addition and scalar multiplication, that this matrix corresponds to a linear transformation of R^n

That a1 and an are alpha-1 and alpha-n.

I really hope this question makes sense to someone. I'm not the strongest at math and I need this course to get into a Master's program. I really hope you can help me!

Thanks so much!
- H.

Edit: I posted this in another form area by accident, and I hope it's ok if I reposted it here. :/
That an operator $H$ is a linear transformation on $\mathbb{R}^n$ means:

$H(\alpha x+\beta y)=\alpha Hx +\beta Hy$

for any $x$ and $y$ in $\mathbb{R}^n$ and $\alpha$ and $\beta$ in $\mathbb{R}$

Here $Hy$ means your matrix is $H$ , $y$ is a (column) vector in $\mathbb{R}^n$ and $Hy$ denote the matrix prioduct of $H$ and $x$.

Now what have you tried, and what problems are you having?

CB

3. Re: Consider this matrix and prove...

I haven't tried too much with this question because I am just stumped at this entire question. I know about transformation matrices and all that, but I wondered if I was supposed to multiply the matrix n x n by x1, x2, x3? I'm just not very strong in this type of math, heh.

4. Re: Consider this matrix and prove...

Originally Posted by hyrule
I haven't tried too much with this question because I am just stumped at this entire question. I know about transformation matrices and all that, but I wondered if I was supposed to multiply the matrix n x n by x1, x2, x3? I'm just not very strong in this type of math, heh.
$Hx=\left[ \begin{array}{cccc}a_1 & 0 &...& 0 \\ 0 & a_2 & ... & 0 \\ \vdots & &\ddots & \vdots \\ 0&...&0&a_n \end{array} \right] \left[ \begin{array}{c} x_1\\x_2 \\ \vdots \\ x_n \end{array} \right]=\left[ \begin{array}{c} a_1 x_1\\a_2 x_2 \\ \vdots \\ a_n x_n \end{array} \right]$