>say pawsome meowster, good job nya~
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What to expect?

@FaustianReactor
Say you have k linear algebraic equations in n variables; in matrix form we write
AX = Y . Give a proof or counterexample for each of the following.
a) If n = k there is always at most one solution.
b) If n > k you can always solve AX = Y .
c) If n > k the nullspace of A has dimension greater than zero.
d) If n < k then for some Y there is no solution of AX = Y .
e) If n < k the only solution of AX = 0 is X = 0.
 
@FaustianReactor
Form 1040 U.S. Individual Income Tax Return 2025 Department of the Treasury—Internal Revenue Service
OMB No. 1545-0074 IRS Use Only—Do not write or staple in this space.
For the year Jan. 1–Dec. 31, 2025, or other tax year beginning , 2025, ending , 20 See separate instructions.
Filed pursuant to section 301.9100-2 Combat zone Deceased MM / DD / YYYY Spouse MM / DD / YYYY
Other
Your first name and middle initial Last name Your social security number
If joint return, spouse’s first name and middle initial Last name Spouse’s social security number
Home address (number and street). If you have a P.O. box, see instructions. Apt. no.
City, town, or post office. If you have a foreign address, also complete spaces below. State ZIP code
Foreign country name Foreign province/state/county Foreign postal code
Check here if your main home, and your
spouse’s if filing a joint return, was in
the U.S. for more than half of 2025.
Presidential Election Campaign
Check here if you, or your spouse if filing jointly, want $3 to go to this fund. Checking a box below will not change your tax or refund.
You Spouse
Filing Status
Check only
one box.
Single
Married filing jointly (even if only one had income)
Married filing separately (MFS). Enter spouse’s SSN above
and full name here:
Head of household (HOH)
Qualifying surviving spouse (QSS)
If you checked the HOH or QSS box, enter the child’s name
if the qualifying person is a child but not your dependent:
If treating a nonresident alien or dual-status alien spouse as a U.S. resident for the entire tax year, check the box and enter their
name (see instructions and attach statement if required):
Digital Assets At any time during 2025, did you: (a) receive (as a reward, award, or payment for property or services); or (b) sell,
exchange, or otherwise dispose of a digital asset (or a financial interest in a digital asset)? (See instructions.) . . Yes No
Dependents
(see instructions)
If more
than four
dependents,
see instructions
and check
here . .
Dependent 1 Dependent 2 Dependent 3 Dependent 4
(1) First name
(2) Last name
(3) SSN
(4) Relationship
(5) Check if lived
with you more
than half of 2025
(a) Yes
(b) And in the U.S.
(a) Yes
(b) And in the U.S.
(a) Yes
(b) And in the U.S.
(a) Yes
(b) And in the U.S.
(6) Check if Full-time
student Permanently
and totally
disabled
Full-time
student Permanently
and totally
disabled
Full-time
student Permanently
and totally
disabled
Full-time
student Permanently
and totally
disabled
(7) Credits Child tax
credit
Credit for
other
dependents
Child tax
credit
Credit for
other
dependents
Child tax
credit
Credit for
other
dependents
Child tax
credit
Credit for
other
dependents
Check if your filing status is MFS or HOH and you lived apart from your spouse for the last 6 months of 2025, or you are legally
separated according to your state law under a written separation agreement or a decree of separate maintenance and you did not
live in the same household as your spouse at the end of 2025.
Income
Attach Form(s)
W-2 here. Also
attach Forms
W-2G and
1099-R if tax
was withheld.
If you did not
get a Form
W-2, see
instructions.
1 a Total amount from Form(s) W-2, box 1 (see instructions) . . . . . . . . . . . . . 1a
b Household employee wages not reported on Form(s) W-2 . . . . . . . . . . . . . 1b
c Tip income not reported on line 1a (see instructions) . . . . . . . . . . . . . . 1c
d Medicaid waiver payments not reported on Form(s) W-2 (see instructions) . . . . . . . . 1d
e Taxable dependent care benefits from Form 2441, line 26 . . . . . . . . . . . . 1e
f Employer-provided adoption benefits from Form 8839, line 31 . . . . . . . . . . . 1f
g Wages from Form 8919, line 6 . . . . . . . . . . . . . . . . . . . . . 1g
h Other earned income (see instructions). Enter type and amount: 1h
i Nontaxable combat pay election (see instructions) . . . . . . . 1i
z Add lines 1a through 1h . . . . . . . . . . . . . . . . . . . . . . 1z
Attach Sch. B
if required.
2a Tax-exempt interest . . . 2a b Taxable interest . . . . . 2b
3a Qualified dividends . . . 3a b Ordinary dividends . . . . . 3b
c Check if your child’s dividends are included in 1 Line 3a 2 Line 3b
4a IRA distributions . . . . 4a b Taxable amount . . . . . . 4b
c Check if (see instructions) . . . . . 1 Rollover 2 QCD 3
5a Pensions and annuities . . 5a b Taxable amount . . . . . . 5b
c Check if (see instructions) . . . . . 1 Rollover 2 PSO 3
6a Social security benefits . . 6a b Taxable amount . . . . . . 6b
c If you elect to use the lump-sum election method, check here (see instructions) . . . . .
d If you are married filing separately and lived apart from your spouse the entire year (see inst.), check here
7a Capital gain or (loss). Attach Schedule D if required . . . . . . . . . . . . . . 7a
b Check if: Schedule D not required Includes child’s capital gain or (loss)
8 Additional income from Schedule 1, line 10 . . . . . . . . . . . . . . . . . 8
9 Add lines 1z, 2b, 3b, 4b, 5b, 6b, 7a, and 8. This is your total income . . . . . . . . . 9
10 Adjustments to income from Schedule 1, line 26 . . . . . . . . . . . . . . . 10
11a Subtract line 10 from line 9. This is your adjusted gross income . . . . . . . . . . 11a
For Disclosure, Privacy Act, and Paperwork Reduction Act Notice, see separate instructions. Cat. No. 11320B Form 1040 (2025) Created 9/5/25
 
Time, @FaustianReactor? Is it really that time again? It seems as if you only just arrived. You've done a great deal in a small time-span. You've done so well, in fact, that I've received some interesting offers for your services.

Ordinarily, I wouldn't contemplate them but these are extraordinary times.

Rather than offer you the illusion of free choice, I will take the liberty of choosing for you... if and when your time comes round again. I do apologize for what must seem to you an arbitrary imposition, @FaustianReactor . I trust it will all make sense to you in the course of... well... I'm really not at liberty to say. In the meantime... this is where I get off.
 
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