Linear combinations and span | Vectors and spaces | Linear Algebra | Khan Academy

Understanding linear combinations and spans of vectors

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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

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[jennahaines]

I have been watching Khan since I was in middle school for math help. Now, I'm at Georgia Tech and STILL getting A's and B's in math because of this man. All my gratitude. You deserve highest honors for teaching the world.

[kyledomen]

This quote really helped me understand the concept.

"If vector a and vector b spans R2, that tells us any vector in R2 can be represented by a linear combination of vector a and vector b"

[adrianchivas]

Thank you khan for helping me understand this. I am taking Linear Algebra my sophomore year in college. Anyways, all I have to say is that you are the real MVP!!!
Always, remember that you are changing lives and I am in college because of you my friend. On track to become an Electrical Engineer.
On behalf of everyone that admires you, Thank you!!!

[mrjoni]

3:47-3:52 " Hey Sal, why are you even introducing this idea of an linear combination?"
"Beacause I wanna!"

[WalidFeghali]

I passed Linear Algebra because of these videos. You're the best, Khan!

[jappmayo]

I just paused this halfway to say THANK YOU for helping me to understand this!

[nlnrose]

... you explain this better than my college matrix algebra teacher... wow I actually am starting to understand this.

[alechenderson]

shoutout to anyone else with a bad linear algebra prof

[TheBlackZoe]

Using this "pseudo-proof" system: drawing the vectors as a visual method is such a great representation of the idea of vector space. Thank you. Greetings from Greece.

[EwoudCP]

Wow. This is really useful. I've been trying to make sense of the idea of the span, but I tend to see things other people don't. More frustratingly, I tend to not see things that other people do. Thanks for the video. I'll definitely keep your playlists in my bookmarks.

[eleazararce]

About to take this class for a second times after 13 years of being out of school. Its should be fun and interesting this time around!! I will be utilizing these videos too!!

[bobbypesch]

vibing in my dorm rn but had to give a shoutout to Sal, this man got me into college

[danielcohen]

How do you know so much? You know math, accounting, business, computers, physics and more.

[selingermann]

I have finally reached ENLIGHTENMENT!
Thank you.

[mabeklvista]

why is this 10 times better explained than in lectures? they go so fast, here i can learn on my own speed. Thank you Sal!

[Zurh]

They've explained me this for 3 consecutive years. Every year I fear this explanation because I can't picture it, I don't know why is it useful, and now it's really like I can see the truth hahaha.
How can you explain everything so nicely in every single video and my teachers be so useless? XD

[krose]

i gave up and just started watching the lectures DURING class!!

[rowanfrankenbach]

7:57 the "SPAN" he said it with gusto lmao. sal is legendary

[roxymang]

so helpful! i was stuck on a question all day reading the textbook, then i watched this video and now it's clear :D

[nathandrouillard]

Thank you. These concepts aren't even that hard, but my prof speaks in VERY broken English and teaches very poorly on top of that so I absorb close to nothing in my lectures. These videos are a godsend