Ultimate Crash Course PDF
Student: If you want a hardcopy, I only print them in small batched. You can be notified of the release via YouTube.com/@authorjond (I do one batch a month usually)
After being a professional tutor and student for about 15 years, I have isolated everything a STEM major needs to know to be successful both inside and outside of college. The book comes with an accompanying YouTube channel and podcast that provide explanations beyond the text.
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Construct a Matrix with Given Column Space and Nullspace (Step-by-Step Proof)
Title: Construct a Matrix with Given Column Space and Nullspace (Step-by-Step Proof)
In this walkthrough, we show how to construct a matrix whose column space contains (1,1,1) and whose nullspace is the line of multiples of (1,1,1,1).
Step 1: Understand the Problem
We want:
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Column space to contain the vector (1,1,1)
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Nullspace to be the set of all scalar multiples of (1,1,1,1)
This means we are looking for a matrix A such that:
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A * x = 0 has solution space spanned by (1,1,1,1)
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The matrix A has 3 rows (since the column vector has 3 entries)
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The matrix A has 4 columns (since the nullspace vector has 4 components)
Step 2: Definitions
Column Space: All linear combinations of the columns
Nullspace: All solutions to A * x = 0
So x = t(1,1,1,1) is the general solution of the homogeneous system.
Step 3: Deconstruct and Solve
We assume:
x = (1,1,1,1)t
So, x₁ = x₂ = x₃ = x₄
This gives:
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x₁ - x₂ = 0
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x₂ - x₃ = 0
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x₃ - x₄ = 0
Which gives us a system of equations:
x₁ - x₂ = 0
x₂ - x₃ = 0
x₃ - x₄ = 0
Written in matrix form:
[ 1 -1 0 0 ]
[ 0 1 -1 0 ]
[ 0 0 1 -1 ]
Thus, a correct matrix A is:
A = [ 1 0 0 -1
0 1 0 -1
0 0 1 -1 ]
This has:
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Column space containing (1,1,1)
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Nullspace spanned by (1,1,1,1)
Step 4: Validation
Any vector in the nullspace must satisfy A * x = 0.
Let x = (1,1,1,1), then:
A * x = [1 -1 0 0]·x = 1 - 1 = 0
[0 1 -1 0]·x = 1 - 1 = 0
[0 0 1 -1]·x = 1 - 1 = 0
✅ Valid
And since (1,1,1) can be expressed as a linear combination of the columns, it's in the column space.
Conclusion
One valid matrix is:
A = [ 1 0 0 -1
0 1 0 -1
0 0 1 -1 ]
This matches the conditions exactly. There are infinitely many matrices that can satisfy these conditions, but this form is simplest and easily validated.
—
This lesson aligns with Chapter 3.2 in Strang’s Linear Algebra (4th ed.). Always cite sources and avoid shortcut tools like AI or uncredited platforms during exams unless explicitly allowed.
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Before Starting Algebra or Calculus Based Physics
Before Starting Algebra or Calculus Based Physics
Unlock your full potential in physics with Before Starting Algebra or Calculus Based Physics — the ultimate prep guide for students stepping into the world of formulas, forces, and functions. Whether you're about to take high school physics, college algebra-based physics, or dive into calculus-based mechanics, this book gives you the exact foundation you need to thrive.
Inside, you’ll master the essential math skills, units, vectors, and logic that most students struggle with when physics begins. We break it all down in simple, clear language with step-by-step examples that connect math concepts to real-world physical scenarios. This isn’t a textbook — it’s a launchpad.
Perfect for freshmen, returning students, or anyone who wants to walk into physics class with confidence.
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You’re intimidated by physics but want to succeed
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The Ultimate Cheat Sheet for Math and Physics
"The Ultimate Cheat Sheet for Math and Physics" is a comprehensive guide written by renowned mathematician and physicist gone fiction author, Jonathan David. This indispensable resource is designed to equip students with a powerful arsenal of tips, tricks, and essential knowledge needed to excel in mathematics and physics. Whether you're studying algebra, trigonometry, precalculus, or diving into the intricacies of differential calculus (Calc 1), integral calculus (Calc 2), several variable calculus (Calc 3), multivariable and vector calculus, linear algebra, differential equations, or physics, this book has got you covered.
Crash Course Hardcover with PDF Access
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[The first edition of this book is currently being printed. Orders will ship once copies arrive—this may take a few weeks. In the meantime, you’ll have full PDF access to this title and all others.]
After 15 years as a student and professional tutor, I’ve distilled everything a STEM major needs to succeed both in and out of college. This book is supported by a YouTube channel and podcast with additional explanations and walkthroughs.
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Note: The “Ultimate Cheat Sheet” is included as the second half of this book.
With the purchase of this title, you will recieve an access code to AuthorJonathanDavid.com where you can download the PDF versions and access all content. Please email me after purchase for this code if you don't want to wait for it to come in the mail. Thanks!
[The first edition of this book is being printed. Orders will be shipped when they arrive. May take a few weeks. You will receive PDF access to this title and all others meanwhile]
After being a professional tutor and student for about 15 years, I have isolated everything a STEM major needs to know to be successful both inside and outside of college. The book comes with an accompanying YouTube channel and podcast that provide explanations beyond the text.
BONUS
Inludes lifetime access code for ad-free videos lessons, podcasts, and audiobooks (limited time offer).
A copy (with matching laminated bookmark) of "i no i'm write" podcasts summariztion 2024.
NOTE* "Ultimate Cheat Sheet" will come as the second half of the book.
Included with "Triple Play"
"i no i'm write" (Volume I, 2024) is a transformative guide by author Jonathan David, summarizing insights from 200 of his YouTube podcasts. Designed to support college STEM majors and anyone feeling adrift, the book introduces "Truth Acknowledgment"—a practice for confronting and embracing life’s realities with clarity and purpose. Blending practical advice, philosophical musings, and real-world applications, this volume offers readers a chance to align their personal and academic paths, rediscover their goals, and navigate life with confidence and resilience.
Included with Membership: "i no i'm write" podcast
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Physics: Electromagnetism
Physics: Mechanics
When writing by hand, just be neat and clear, as bold and italicization are not too easy by hand. When doing homework, you should be typing professional-looking documents, whereas when doing exams—that is, practice exams- you should practice by writing with pen or pencil and in a very unpleasant situation to mimic the real exam. Take away: When it comes to units, students lose a lot of points for doing something like finalizing the answer by saying 10. Rather than 10.0 m/s. You will also have to account for significant figures. Meaning, 10 may be wrong and 10.0 is correct, but still wrong if not adding the units. A physics student can do all the calculations correctly and still get a bad grade simply for sloppy notation. NASA would not hire someone who is not attentive to detail, nor would Apple, Google, or any other huge company that pays a lot of money for precision and accuracy! As you begin your journey in physics, if you have not already restructured your way of thinking to be a professional, now is the time to do so!
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