linear algebra – Showing vector is in range of matrix

We have the matrices $A in mathbb R^{n times d}, W_0 in mathbb R^{d times d}$ where $d geq n$ and $rank(A) = n, rank(W_0) = d$.

$x in mathbb R^{d times 1}$ is a vector in range of $W_0^TA^T$. I want to show that it’s also in range of $W_1^TA^T$ where

$W_1 = W_0 -alpha A^TAW_0xx^T$ where $alpha in mathbb R$

What I tried:

We know $x = W_0^TA^Tv_0$, we want to find $v_1$ such that $W_1^TA^Tv_1 = W_0^TA^Tv_0$

In other words, $(W_0^T-alpha xx^TW_0^TA^TA)A^Tv_1 = W_0^TA^Tv_0$

Consider $W_0^T-alpha xx^TW_0^TA^TA$. the term on the right is a rank $1$ matrix, so by the matrix determinant lemma $det(W_0^T-alpha xx^TW_0^TA^TA) = (1 – alpha x^TW_0^TA^TAW_0^{T^{-1}}x)det(W_0^T)$

We know $det(W_0^T) neq 0$. If we assume that $alpha$ is such that the left term is non zero (we can assume such a thing) then $W_0^T-alpha xx^TW_0^TA^TA$ is invertible, so we get

$A^Tv_1 = (W_0^T-alpha xx^TW_0^TA^TA)^{-1}W_0^TA^Tv_0$

Now $A^T$ is a $d$ by $n$ matrix of rank $n$ so it has a left inverse, we can multiply by it and get:

$v_1 = A^{T^+}(W_0^T-alpha xx^TW_0^TA^TA)^{-1}W_0^TA^Tv_0$

The problem is that this operation is not invertible. I may have gotten an expression for $v_1$ but there’s no gaurantee that this $v_1$ is such that $W_1^TA^Tv_1 = x$.

Any ideas?

linear algebra – Showing the orthogonal subspace to the Range of A is the null space of the complex conjugate of A

I have matrices $A in mathbb{C}^{m times n}$

My goal is to show that $R^{perp}(A) = N(A’)$.

Conceptually I get that the orthogonal space to the range of the matrix is going to be the null space because otherwise it’d be in the row space but I’m having a hard time expressing it in math.

I haven’t done too much complex matrix work so I’m not sure what pitfalls I need to be wary of

Q: How can I find the range of this function?

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I’m just starting to learn about functions, and I have no idea how to calculate the range of a function both radical and rational

wide angle – Best mid to high range camera nowadays to use for shooting big farm equipment

I am a graphic designer and I do a bit of photography but not on an expert level so I thought I’d ask here for I’m sure I will get a lot of insights. What is the best mid to high range cameras nowadays for shooting big farm equipment in the field? And what lenses as well? Also I am deciding between DSLR and mirrorless. The main purpose of the camera is for taking pictures, and video is secondary but it would be nice to have the capability of both still photo and video.

I provided an image to show what kind of pictures I am taking. Keep in mind though I do not own the photo. Thanks everyone!

enter image description here

macos – How to include footer cells in formula when selecting column range?

I have a sheet in Numbers where I have couple of values in columns and have one footer row. I need to include the cell in column including the footer. For e.g. in the below screenshot, the XIRR formula needs to include the cells A6 and B6 when using formula as Payments and Date respectively. Is it possible to achieve it?

enter image description here

linux – ext4 restore backup: deleteino: out of range 0

I’m trying to use (dump) restore -command to restore linux ext4 file system from an incremental backup. I’m using a hanoi tower strategy to make the backups, and the dumps look like as follows, sorted by time:

root@server:/backup# ls -alth
-rw-r--r--  1 backup backup  264305457 Oct  2 22:05 l2-phz-xvda2-001
-rw-r--r--  1 backup backup  755017563 Nov  3 22:13 l1-phz-xvda2-001
-rw-r--r--  1 backup backup    1734399 Nov 11 22:00 l4-phz-xvda2-001
-rw-r--r--  1 backup backup  498703367 Nov 19 22:08 l3-phz-xvda2-001
-rw-r--r--  1 backup backup 2304401315 Jan  1 22:47 l0-phz-xvda2-001
-rw-r--r--  1 backup backup     933481 Jan  5 22:00 l5-phz-xvda2-001
-rw-r--r--  1 backup backup  122383278 Jan  6 22:02 l7-phz-xvda2-001
-rw-r--r--  1 backup backup  122803554 Jan  7 22:02 l6-phz-xvda2-001

Since the level 0 -dump is a full dump, I first restored that, but when I try to restore the next (l5) dump, I get the following error message:

restore -rMf /backup/l5-phz-xvda2- -b 256
Dump tape is compressed.
deleteino: out of range 0
abort? (yn) n
deleteino: 0 not found
abort? (yn) n
deleteino: out of range 0
abort? (yn) y
dump core? (yn) y
Aborted (core dumped)

Any suggestions how to restore the incremental dumps?

integration – dividing a range of randam variable into 3 equal or unequal parts.

This is my first post on Math-Exchange. I’m a student, and I work with python, but when things gets too “algebraic” I get stuck. So I thought I’d ask over here.

I need to create an algorithm that divides a range of random variables into 3 parts ( equal or unequal) by using the highest number of that range and the lowest number of that range.
What is the technique for that?

Suppose the age of n number of peoples 20,30,6,90, 54,98,83,33, 24,19,56,74,88,90,43,45,74,50,76 and so on. Here, highest number is 98 and the lowest number is 6. now i want to divide that range into 3 parts (equal or unequal) by using that highest and lowest number.

problem /24 range ips

we have a range ip from ripe
we set correctly rdns in ripe and everything is ok
but suddenly from 1 month ago all ips in this range… | Read the rest of

sql server – Double range query on one column uses wrong seek predicate

This problem comes up in the real world, but is already present in the following trivial example. I am using SQL Server 2017.

Given a table with only one non-id column, which takes only one value:

create table #test (id bigint not null, status smallint not null)

/* Populate with 10000 times value 10 */
;WITH numbers(Number) AS
(SELECT 1 AS Number
 SELECT Number+1 FROM numbers where Number<10000

insert into #test (id,status)
select number,10 from numbers option(maxrecursion 10000)

/* Create fresh index */
create index index_status on #test ((status))
DBCC SHOW_STATISTICS ("tempdb..#test", 'index_status') WITH HISTOGRAM   

Now we query a double range on this column (0 results, since the only value is outside both ranges):

select 1
from #test
where status < 10
and status <> 2

The chosen execution plan uses an Index Seek, with

  • Seek Predicate “status < 2 AND status > 2”
  • Predicate “status < 10”

and consequently needs to read all 10000 rows. This surprises me, since I can read off the index statistics to conclude that a Seek Predicate on “status < 10” would be more efficient, since this already filters out all rows.

Question: Why are Seek Predicate and Predicate chosen ‘the wrong way’?

I am not interested in alternatives or workarounds, I am only interested in understanding this choice of execution plan. For example we can rewrite the query using a UNION ALL on both range queries seperately, forcing two efficient Seek Predicates:

select 1
from #test
where status < 2

union all 

select 1
from #test
where status > 2 and status < 10

pathfinder 1e – Can blank target and range spells be considered valid for the teamwork feat share spells?

Unfortunately, as limiting as this seems, Share Spells specifically only works with spells with a Target entry of You. Lacking a Target entry is not the same thing as a Target entry of You.

The Range entry of a spell also has nothing to do with whether or not it will work with Share Spells. As an example, the Lead Blades spell has Range Personal, but the Target is Touch rather than You. Therefore, it would not work with Share Spells.

Neither of the examples you listed would work with Share Spells, even though they possibly seem like something that could. Both Bless and Detect Evil have an Area entry rather than a Target.