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Linear AlgebraLecture 1: Vector Spaces

ENGAGELinear Algebra


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NavigationVector Spaces – Intuition Video

Imagine standing on one of the floors of a tall office building. In your pocket are pieces of paper with instructions on them.




One example might be... Take “one step forward”




One example might be... Take “one step forward”

Another might be “walk sideways two steps to the left”



Now, you are also allowed to combine your instructions. You are allowed to add them.. for example you might...

E.g. take “one step forward” & “walk sideways two steps to the left” i.e.



Now, you are also allowed to combine your instructions. You are allowed to add them.. for example you might...

E.g. take “one step forward” & “walk sideways two steps to the left” i.e.

You could also combine the instructions by doing one, multiple times (i.e. you could multiply)

E.g. take “one step forward” twice. i.e.



You could also do more advanced things with addition and multiplication.

You are also allowed to take negative multiples... E.g. take “one step forward” -1 times. I.e. take one step

backwards



You could also do more advanced things with addition and multiplication.

You are also allowed to take negative multiples... E.g. take “one step forward” -1 times. I.e. take one step

backwards.

You are also allowed to take non-integer multiples... E.g. take “one step forward” 4/3 times. I.e. take one step

and 1/3 of a step forward.

One step forward vs. 4/3 steps forward.



What can we do now?

Go back to the office building and start to move around..



With just one instruction... Let’s say it’s walk “one step to the right” you can now walk anyway along a line...




With two instruction we can go further.... Let’s say it’s walk “one step to the right” but also take “one step forward.”

You can now see how you would be able to walk everywhere in the whole floor.




With two instruction we can go further.... Let’s say it’s walk “one step to the right” but also take “one step forward.”

You can now see how you would be able to walk everywhere in the whole floor.

Add one more instruction.... Let’s say take the lift up one floor (given we can add and multiply our instructions. We can now go everywhere in the entire building!



So what have we discovered?? We’ve found that with some instructions, and some rules for

combining those instructions we can move along say a line, or maybe an entire floor, or maybe even an entire office block.





This set of lectures is on vector spaces, and to give you some intuition to three new concepts coming up in the next lecture...

1. Vectors are like.... the instructions (e.g. take “one step forward”).







2. Vector addition and scalar multiplication are like.... the combining rules for our instructions (i.e. adding and multiplying).







2. Vector addition and scalar multiplication are like.... the combining rules for our instructions (i.e. adding and multiplying).

3. Vector spaces are like.... all the places you could get to given the instructions you had. So one ‘vector space’ was a line, another was the entire floor and another might be the entire building. It’s basically everywhere you can travel to given the instructions (and the rules of vector addition and scalar multiplication) you have.



Now you have an insight into what vector spaces are, the remaining part of the lecture shall be aimed at building intuition as to the mathematical definition of a vector space.



So we have seen, vector spaces, are ‘all the places you can get to’ given a set of directions that you have.

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