Fun linear algebra exercise. (1 Viewer)

seanieg89 · Mar 15, 2014

Motivated by Sy's last question, here is a fun problem for those with some linear algebra knowledge:

$1. If $T:X\rightarrow Y$ is a linear map between vector spaces, and for some $q\in Y$, $p\in X$ is a solution to the equation $Tx=q$, prove that the general solution is given by $x=p+r$ where $r\in \ker(T)$.\\ \\ 2. What is the general solution of the equation $\left(\frac{d}{dx}-\lambda\right)u=0$ where $\lambda\in \mathbb{C}$ and $u\in \mathcal{C}^\infty(\mathbb{R})$?\\ \\ 3. Explain why the general solution to a linear ordinary differential equation with constant coefficients is a subspace of $\mathcal{C}^\infty(\mathbb{R})$ of dimension $n$. Find a basis for this subspace.$

braintic · Mar 16, 2014

seanieg89 said:
Motivated by Sy's last question, here is a fun problem for those with some linear algebra knowledge:

$1. If $T:X\rightarrow Y$ is a linear map between vector spaces, and for some $q\in Y$, $p\in X$ is a solution to the equation $Tx=q$, prove that the general solution is given by $x=p+r$ where $r\in \ker(T)$.\\ \\ 2. What is the general solution of the equation $\left(\frac{d}{dx}-\lambda\right)u=0$ where $\lambda\in \mathbb{C}$ and $u\in \mathcal{C}^\infty(\mathbb{R})$?\\ \\ 3. Explain why the general solution to a linear ordinary differential equation with constant coefficients is a subspace of $\mathcal{C}^\infty(\mathbb{R})$ of dimension $n$. Find a basis for this subspace.$

I take it this is not first year linear algebra?

seanieg89 · Mar 16, 2014

braintic said:
I take it this is not first year linear algebra?

I think it is definitely be within the scope of at least some first year linear algebra courses. They vary pretty widely in content and flavour.

anomalousdecay · Mar 16, 2014

brb in 3 months when I've done all the matrices and vectors stuff in detail.

Fun linear algebra exercise. (1 Viewer)

seanieg89

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braintic

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seanieg89

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anomalousdecay

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