Adv. Linear Algebra

[jddenham]

Hello all, I am new to the forum and having some difficulty with my Linear Algebra class. I am having a hard time with learning all the terms in this course and would like some decent explanation and examples. I have read the chapters and went to the internet for additional help, but I am still not completely clear on the topics. If anyone has any information or advice I would be very much appreciative.

Vector space: if we have a set of vectors {v1, v2,....,vn} then the vector space is the collection of those vectors?

subspace: If V is a vector space and W is a subset of V. Then W is a subspace of V iff W is closed under addition, scalar multiplication, has a zero vector, and each vector in W has an additive inverse in W.

span of (S): the set of all linear combinations of the vectors in S

spans: a subset of a vector space spans the vector space if the span of the subset equals the vector space?

basis: a linearly independent subset of a vector space that spans the vector space?

dimension: number of vectors in each basis of a vector space?

null space of: set of all vectors x for which Ax=0?

range of: ?

Thanks in advance for your help.

 

[tkhunny]

"Vector Space" is no good. It's not just a collection. Start with the propoerties listed in "SubSpace".

 

[burakaltr]

Hello all, I am new to the forum and having some difficulty with my Linear Algebra class. I am having a hard time with learning all the terms in this course and would like some decent explanation and examples. I have read the chapters and went to the internet for additional help, but I am still not completely clear on the topics. If anyone has any information or advice I would be very much appreciative.

Vector space: if we have a set of vectors {v1, v2,....,vn} then the vector space is the collection of those vectors?

subspace: If V is a vector space and W is a subset of V. Then W is a subspace of V iff W is closed under addition, scalar multiplication, has a zero vector, and each vector in W has an additive inverse in W.

span of (S): the set of all linear combinations of the vectors in S

spans: a subset of a vector space spans the vector space if the span of the subset equals the vector space?

basis: a linearly independent subset of a vector space that spans the vector space?

dimension: number of vectors in each basis of a vector space?

null space of: set of all vectors x for which Ax=0?

range of: ?

Thanks in advance for your help.

I will try to help you with the best of what I know :

Take the 3-D space, we have i , j , k

i=<1,0,0>
j=<0,1,0>
k=<0,0,1>

if these are orthogonal vector, which they are :

I can write a vector with the Linear Combinations of these unique perpendicular ( to each other ) vectors

I start by forming , let's say <a,b,c> vector

How do i go ?


<a,b,c> = A1*i + A2 * j + A3*k where A1,2,3 are Real constants

so that a maps to A1 ( etc. ) by vector definition

see that for the zero vector <0,0,0> ,

I can form the zero vector if and only if A1,A2,A3 are all zero.

AND

I cannot , " in any linear combination ", get i in terms of j,k ===> i.e can you find solution to i=B1*j+B2*k ? ===> No Real B1 or B2 can satisfy this condition !

I cannot get j in terms of i,k

I cannot get k, in terms of i,j

To be a vector base : (1) all base vectors can come together to point a vector in space;

that is

vector<x,y,z> = L1 * <1,0,0> + L2 * <0,1,0> + L3*<0,0,1>
===> see how it is possible to find reals L1 equaling x and L2 equaling y; and L3 equaling z

If I can get one vector in only one combination that is if i can get <0,0,0> in only one combination of the basis vectors ; only then they are considered to be Linearly Independent.

See that I can only get <0,0,0> when i multiply i with 0;j with 0 And k with zero

That is all my knowledge , I hope someone can further explain.

Hope this will be helpful.

Best Wishes.

 

[burakaltr]

Also 4-D or 5-D vectors are possible :

4D , <x,y,z,t> vector
5D , <x,y,z,t,s>

and n dimensions a vector can have - ie : <d1,d2,d3,d4,d5,d6....,dn>

We can still talk about orthogonality, linear independence and possibility to form the zero vector <0,0,0,0,0,0,0,0....nzeroes> only by setting each dimension's real coeff.s to zero.

 

[jddenham]

Thx

Appreciate the help. Its still hard to understand coming from a non-math major background, but all info helps.