# Consider this matrix and prove...

• Oct 6th 2011, 09:51 PM
hyrule
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. :/
• Oct 6th 2011, 10:53 PM
CaptainBlack
Re: Consider this matrix and prove...
Quote:

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 $\displaystyle H$ is a linear transformation on $\displaystyle \mathbb{R}^n$ means:

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

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

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

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

CB
• Oct 6th 2011, 11:17 PM
hyrule
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.
• Oct 7th 2011, 12:56 AM
CaptainBlack
Re: Consider this matrix and prove...
Quote:

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.

$\displaystyle Hx=\left[ \begin{array}{cccc}a_1 & 0 &...& 0 \\ 0 & a_2 & ... & 0 \\ \vdots & &\ddots & \vdots \\ 0&...&0&a_n \end{array} \right] \left[ \begin{array}{c} x_1\\x_2 \\ \vdots \\ x_n \end{array} \right]=\left[ \begin{array}{c} a_1 x_1\\a_2 x_2 \\ \vdots \\ a_n x_n \end{array} \right]$