linear algebra – Condition for a family of elements of an affine space to be affinely independent

Let $E$ be an affine space attached to a $K$-vector space $T$. For a
family $(x_i)_{iin I}$ of elements of $E$ and any $ain E$, the set
$$left{sum_{iin I}lambda_i(x_i-a)+a | lambdain
K^{(I)}landsum_{iin I}lambda_i=1right}$$
is the affine linear
variety generated by
$(x_i)_{iin I}$.

Let $(a_i)_{iin I}$ be a nonempty family of elements of $E$ and $kin
I$
. This family is said to be affinely independent if and only if the
family $(a_i-a_k)_{ine k}$ is linearly independent in $T$.

Now, suppose $(a_i)_{iin I}$ be a nonempty family of elements of $E$. Suppose that, for every $kin K$, $a_k$ doesn’t not belong to the affine variety generated by $(a_i)_{ine k}$. I want to show that this implies that $(a_i)_{iin I}$ is affinely independent.

Attempt:

Since $Ineemptyset$, let $kin I$. According to the above, it is sufficient to show that the family $(a_i-a_k)_{ine k}$ is linearly independent in $T$. So, let $lambdain K^{(I-{k})}$ such that $sum_{ine k}lambda_i(a_i-a_k)=0$. Then $sum_{ine k}lambda_i(a_i-a_k)+a_k=a_k$. This implies that $sum_{ine k}lambda_ine 1$. But how can this help me show that $lambda=0$?

backup – catalina, IMAc – Under volumes – folders which has taken storage space

suddenly today I saw my storage space taken over by a folder of over 200gb.. I have an iamc 2017 ruuning on 10.15.7 , 1 tb ssd.
Under Macintosh/volumes there is a folder named after my External HDD under which I find folders, USERS, Applications, Library…which are all duplicated folders…these occupy 239 gb and seem to be rad only which shows up in Daisy disk…Due to this my disk space has been eaten up..WHen i take backup on an external disk using Carbon copy cloner this is not backed up..How to get over this problem?

Linux VM showing incorrect Space in OS

I have create one VM guest (on esxi) , OS Ubuntu 16 server. I assigned it 900 GB but I am seeing only 200 GB of disk space.

root@XXX-log:/temp# df -h
Filesystem                         Size  Used Avail Use% Mounted on
udev                               7.8G     0  7.8G   0% /dev
tmpfs                              1.6G  900K  1.6G   1% /run
/dev/mapper/ubuntu--vg-ubuntu--lv  196G   92G   94G  50% /
tmpfs                              7.9G  8.0K  7.9G   1% /dev/shm
tmpfs                              5.0M     0  5.0M   0% /run/lock
tmpfs                              7.9G     0  7.9G   0% /sys/fs/cgroup
/dev/sda2                          976M  146M  764M  16% /boot
tmpfs                              1.6G     0  1.6G   0% /run/user/0
root@XXX-log:/temp#
root@XXX-log:/temp#
root@XXX-log:/temp#
root@XXX-log:/temp# fdisk -l
Disk /dev/sda: 920 GiB, 987842478080 bytes, 1929379840 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: gpt
Disk identifier: 2C824CE0-94E5-4515-B28D-8FA40983CFF5

Device       Start        End    Sectors  Size Type
/dev/sda1     2048       4095       2048    1M BIOS boot
/dev/sda2     4096    2101247    2097152    1G Linux filesystem
/dev/sda3  2101248 1929377791 1927276544  919G Linux filesystem


Disk /dev/mapper/ubuntu--vg-ubuntu--lv: 200 GiB, 214748364800 bytes, 419430400 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes

What seems wrong here & how to rectify? Kindly advise

storage – Unisphere dynamic pools. Does this mean I can lose more than 1 disk with a raid 5 configuration and 20 drives or how are spare space extents treated?

So emc has implemented dynamic pools which do away with hot spares and instead uses some drive extents as spare space on each drive.

When a drive fails, the contents are copied into this spare space (is it usable living on this spare space?).

When setting up a traditional system, you would normally have a couple hot spares so if a drive fails, it rebuilds onto these spares thus saving you a failure. (Raid 5 with a couple hot spares allows you to lose 3 drives at seperate times i believe).

With this dynamic pool, how does this work? If I have a 20 drive system and use raid 5 on a dynamic pool…what number of drives can I lose (at different times of course) and still be operational?

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Formatted my sd card as internal storage in my Xiamoi phone. but still getting no enough space message when installing apps

I have a Xiamoi note 8 phone running android 9. It has 32gb internal storage and I used most of it, so I decided to buy an sd card to use as internal storage.
I went in settings chose the format as internal storage option. It then asked me to move data to the new internal storage, I clicked yes then after it finished I restarted my phone and It still was giving the low on storage message.

when I open the file manager it shows the phone’s data split evenly between the sd card and the storage.

when I open the settings and go to storage it only shows the phone’s original 32gb.
I searched a lot but couldn’t find a solid solution for this problem.

unity – Looping world for a top down space shooter?

I am trying to understand how to make a seamless looping world for my top down space shooter.
First thing I tried was using multiple cameras, and teleporting the player when he reaches the edge, but it is not working well, and I don’t like that I have to use 4 cameras.
I wanted to test making the objects in the world teleport around the player, but I will have problems with the fact that each object is a different size, and teleporting them around will for sure create problems with moving objects. (Big obstacles could be teleported away and not collide with, for example, bullets until the bullet gets teleported too but inside the big obstacle)
Are there other ways to accomplish this?

How to prove that a tychonoff space $C(X, Y)$ equipped with fine topology can be embedded in a hilbert cube.

As we know that every tychonoff space can be embedded in a hilbert cube. We have that a function space $C(X, Y)$ where $X$ is a tychonoff space and $Y$ also a tychonoff metric space, is a tychonoff space. How can we then prove that it can be embedded in a hilbert cube?