Vector Spaces - Null Spaces, Column Spaces, and Linear Transformations
4 important questions on Vector Spaces - Null Spaces, Column Spaces, and Linear Transformations
The null space of an m x n matrix A, written as Nul(A), is?
Nul A = {x : x is in Rn and Ax = 0}
Theorem 2
The null space of an m x n matrix A is a subspace of Rn. What set of solutions is also a subspace of Rn?
The column space of an m x n matrix A, written as Col A, is?
Col A = Span{a1, ..., an}
NOTE THE SPAN
- Higher grades + faster learning
- Never study anything twice
- 100% sure, 100% understanding
For my 1st exam I got a C. Then, I started looking for ways to learn better and I came across Study Smart. Since then I have scored a A+, A, A- and another A. I highly recommend it to everyone who wants to hear it. I am so happy with it!!
I was struggling to finish all my first-year subjects for 3 years. Then I discovered Study Smart, which helped me to finish all of them within 3 months.
Because of StudySmart, things are going well in many ways. Before, I always had study stress and no time for anything. Now I feel calm and confident (because of the program). I used to score a B, now I usually score an A or A+.
Hey Chris, I really want to thank you very much for developing the platform, I usually got around C for my exams, now I scored an A+, an A and a A- !!
Since using Study Smart, my average has increased significantly. I even completed a statistics course with a A+ while I'm bad at statistics. I took on extra courses this year, because of Study Smart. Thanks so much!
Ever since I started studying with StudySmart I passed all my exams !!! No more re-sits!! At first, I was happy with a pass and now I only get good grades. Many thanks!
I aced all the courses I studied with Study Smart. For law, I first scored a D, and now an A! I’m 100% sure that I’ll pass all the courses I study with Study Smart.
I always thought I studied well. My grades were fine. I could never have imagined that applying these study skills would have such a huge and direct impact on my grades, my speed and the pleasure I have in learning.
For the first time, I finally feel totally confident about the way I learn. I have the perfect overview and make great progress in terms of speed and the grades I get. Thanks, Chris!
I immediately started to enjoy learning again, and my grades soared. All of a sudden, everything went better. I highly recommend Study Smart to all students..
Being able to create my materials online made the process of studying so much smoother and quicker. I no longer have to spend a huge amount of my time on creating my summaries.
At first, my notes were a mess. Now I use Study Smart to take my notes in a perfectly structured way. It helps me to save a lot of time and do other things I enjoy.
Students that study with Study Smart get better grades. The system applies all the learning methods in such a way that both well-performing students and students who struggle perform much better.
Hi Chris, I would like to thank you very much. Last year I scored C on a course, and this year I got an A+ thanks to your method! Thank you very much!
The largest effect is that my child is more enthusiastic about school! Every parent wants their children to be happy at school, Study Smart made it happen.
In high school, I failed several exams, and that has really changed now. I actually needed this tool very much back then. Since I started studying with StudySmart, I haven't had any D's. My average score is now an A and those are results I would have never had in the past
With my old method, I passed only 3 out of 8 courses. Since I started taking my notes digitally in Study Smart, I have passed all my courses on the first try. For me, StudySmart takes away the stress of wondering if I will pass or not.
Thanks to StudySmart, I passed all my exams, and with better grades than before! On top of that, I have mastered a very good study method now, which I am confident will help me earn my degree.
Theorem 3
The column space of an m x n matrix A is which subspace?
The question on the page originate from the summary of the following study material:
- A unique study and practice tool
- Never study anything twice again
- Get the grades you hope for
- 100% sure, 100% understanding
