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# Vector Subspaces - ?!?! watch

1. im not looking for answers to the questions shown below

just want to know how to do such questions

i know i need to check if:
1. zero vector is an element of the space
2. it is closed under addition
3. closed under scaler multiplication

but i still dont understand this!

and i dont understand how to interpret these geometrically sometimes, i can interpret them when they are obvious, like for S5 it is clearly the z axis line and S2 is a unit sphere but what about the others and harder ones?

THANKS!!
2. I'll focus on S1 here. The same methods are used to evaluate the other subsets However if just one of these conditions doesn't hold then it isn't a subspace and you don't need to consider any of the other conditions. But you do need to show all three conditions hold to prove that a subset is a subspace.

1. You need to check if the vector is in the subspace, in other words, plug the values , and into the condition .
2. Closure under addition is the condition that, given any two vectors in the subspace, their sum also in the subspace. If and are in the subspace, then and . Then does also satisfy the condition?
3. Similar to 2. Does satisfy the condition for any ?

As for geometrical interpretations, try drawing a "3D graph" and plotting the points. I know that isn't straightforward but it might give you some intuition.

Also note that is very similar to . Is the geometrical interpretation clearer if I write = ?
3. (Original post by Dragon)
I'll focus on S1 here. The same methods are used to evaluate the other subsets However if just one of these conditions doesn't hold then it isn't a subspace and you don't need to consider any of the other conditions. But you do need to show all three conditions hold to prove that a subset is a subspace.

1. You need to check if the vector is in the subspace, in other words, plug the values , and into the condition .
2. Closure under addition is the condition that, given any two vectors in the subspace, their sum also in the subspace. If and are in the subspace, then and . Then does also satisfy the condition?
3. Similar to 2. Does satisfy the condition for any ?

As for geometrical interpretations, try drawing a 3D graph and plotting the points. I know that isn't straightforward but it might give you some intuition.

Also note that is very similar to . Is the geometrical interpretation clearer if I write = ?
thanks a lot
this has definitely helped a lot

can i asked one more question:

i dont understand what the question wants me to show
4. Never seen this "lin" notation before, but judging from the context I assume it means span. Do you know anything about bases or spans or linear independence?
5. (Original post by Dragon)
Never seen this &quot;lin&quot; notation before, but judging from the context I assume it means span. Do you know anything about bases or spans or linear independence?
yeah i know a little bit
6. This is a bit tricky as I'm not sure exactly what you've been taught, but I'll give you a couple of hints to point you in the right direction.

Using the question's notation, assuming lin means span, . If for some , what can you say about any vector of the form ? It might help to visualise or draw such vectors in , which is just the standard plane in two dimensions.

For the second if and only if, use the fact that the determinant of a matrix is zero if and only if the columns are linearly dependent. Note this fact applies for square matrices of any size.
7. (Original post by Dragon)
This is a bit tricky as I'm not sure exactly what you've been taught, but I'll give you a couple of hints to point you in the right direction.

Using the question's notation, assuming lin means span, . If for some , what can you say about any vector of the form ? It might help to visualise or draw such vectors in , which is just the standard plane in two dimensions.

For the second if and only if, use the fact that the determinant of a matrix is zero if and only if the columns are linearly dependent. Note this fact applies for square matrices of any size.
hmmm
makes more sense than before..

im hopeless!!
these questions are going to be the death of me
8. (Original post by Milan.)
these questions are going to be the death of me
Death by mathematics! Sounds like a good title for a murder/mystery.

If you've not covered linear algebra / vector spaces before, they can take a bit of getting used to. Just practise, think about what you're doing. Well worth getting familiar and comfortable with as they crop up a lot.
9. (Original post by ghostwalker)
Death by mathematics! Sounds like a good title for a murder/mystery.

If you've not covered linear algebra / vector spaces before, they can take a bit of getting used to. Just practise, think about what you're doing. Well worth getting familiar and comfortable with as they crop up a lot.
we've spent the year doing linear algebra and vector spaces, i understand some of the later parts of my linear algebra course much more than this stuff
im more comfortable with things like
matrix diagonlisation, orthonormal basis, markov chains, complex numbers etc.
its the vectors and subspaces and related things i dont get

not enough time to practise, 4 other modules to understand too :/
im just grateful people on TSR can help and 99% of the time i understand it after!

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Updated: April 15, 2011
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