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. :/

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

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.

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.

Because your matrix is diagonal

$\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] $

CB

Re: Consider this matrix and prove...

Oooh ok, I get it. So because I have alpha-n number of all real numbers in the diagonal matrix, I needed to multiply it with Xn number of variables.

I had a suspiscian that I would have to multiply by x variables, but I just wasn't confident in my own answer. You've really helped me understand it better, thank you!!