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Thread: Elementary linear algebra, can't solve any problems from the subspaces problem set???

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    Elementary linear algebra, can't solve any problems from the subspaces problem set???

    So, it's my very first exposure to higher math. I got As all the way from Calc I-III including Dif eq, but now, I find myself unable to solve ANY linear algebra problem from Gilbert Strang's course on MIT OCW..

    Well, actually I can do problem 6.2 (3.2 # 18 in his book), but I need help with this subspace problem:

    Problem 6.1: (3.1 #30. Introduction to Linear Algebra: Strang) Suppose Sand T are two subspaces of a vector space V.

    a) The sum S + T contains all sums s + t of a vector s in S anda vector t in T. Show that S + T satisfies the requirements (addition andscalar multiplication) for a vector space.

    b) If S and T are lines in R^m, what is the difference between S + T andS T? That union contains all vectors from S and T or both. Explainthis statement: The span of S ∪ T is S + T.

    Also, does anybody have any advice on how to study for this course? The chapters in the book, the lectures, recitation contain absolutely Nothing ​about the union and all that..
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    Junior Member Walagaster's Avatar
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    Re: Elementary linear algebra, can't solve any problems from the subspaces problem se

    Quote Originally Posted by Awesome31312 View Post
    So, it's my very first exposure to higher math. I got As all the way from Calc I-III including Dif eq, but now, I find myself unable to solve ANY linear algebra problem from Gilbert Strang's course on MIT OCW..

    Well, actually I can do problem 6.2 (3.2 # 18 in his book), but I need help with this subspace problem:

    Problem 6.1: (3.1 #30. Introduction to Linear Algebra: Strang) Suppose Sand T are two subspaces of a vector space V.

    a) The sum S + T contains all sums s + t of a vector s in S anda vector t in T. Show that S + T satisfies the requirements (addition andscalar multiplication) for a vector space.

    b) If S and T are lines in R^m, what is the difference between S + T andS T? That union contains all vectors from S and T or both. Explainthis statement: The span of S ∪ T is S + T.

    Also, does anybody have any advice on how to study for this course? The chapters in the book, the lectures, recitation contain absolutely Nothing ​about the union and all that..
    Many students have difficulty with their first course in linear algebra, so don't feel like you are alone. The problem is that you are not used to dealing carefully with definitions and axioms. I will show you how to do one part of your question as an indication of how to proceed. So you are given two subspaces $S$ and $T$ of a vector space $V$. And you are given a set$$
    S+T = \{s + t| s\in S,~t\in T\}$$ and you want to show $S+T$ is a subspace. That means you have to show $S+T$ is closed under addition and scalar multiplication. So what you have to show is that if $u$ and $v$ are in $S+T$, then so are $u+v$ and $cu$ for any scalar $c$. So start by assuming $u,v \in S+T$. That means $u=s_1+t_1$ and $v=s_2+t_2$ where $s_1,~s_2 \in S$ and $s_2,~t_2 \in T$. So let's calculate $u+v$ and see if we can show it is in $S+T$.$$
    u+v =s_1+t_1 +s_2+t_2 = (s_1+s_2) + (t_1+t_2) \in S+T$$ since $s_1+s_2 \in S$ and $t_1+t_2 \in T$ because $S$ and $T$ are subspaces. This argument also uses the properties of addition in the vector space to rearrange the sum. Now, can you do a similar argument to show $cu\in S+T$?
    Last edited by Walagaster; Jun 16th 2018 at 10:43 AM.
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    Re: Elementary linear algebra, can't solve any problems from the subspaces problem se

    To add to what Walagaster said, many students don't realize that math definitions are working definitions. That is, you use the exact words of the definitions in proofs. Here the basic problem is to "show that S+ T satisfies the requirements for a vector space". Here "requirements" are the parts of the definition of "vector space". Those parts are (they are presumably in your text) that a vector space consists of a set of objects (vectors) together with two operations ("+" and "*", addition and scalar multiplication) such that if u, v, and w are vectors and a is a number then
    1)u+ v= v+ u (addition of vectors is commutative)
    2)(u+ v)+ w= u+ (v+ w) (associativity of vector addition)
    3)There exist a vector 0 such that u+ 0= u for all vectors, u. (vector identity)
    4)For any vector, u, there exist a vector v such that u+v= 0. (additive inverse. That particular "v" is usually written "-u")
    5)For any numbers, a and b, a*(b*u)= (ab)*u. (associativity of scalar multiplication)
    6)For any numbers, a and b, (a+ b)*u= a*u+ b*u (distributivity of scalar multiplication)
    7)For any number, a, and vectors u and v, a(u+ v)= au+ av (distributivity of scalar multiplication over vector sums)
    8)For any vector, u, 1*u= u. (1 is the multiplicative identity for scalar multiplication)
    (You can see these at Vector Space -- from Wolfram MathWorld)

    To show that T+ S is itself a vector space you must show that all 8 of those are true for vectors in T+ S. For example, to show 1, if u and v are vectors in T+ S then we can write u= p+ q and v= r+ s where p and r are vectors in T and q and s are vectors in S. Then
    u+ v= (p+ q)+ (r+ s)= (p+ r)+ (q+ s) (because 1 is true for vector space V), so
    = (q+ s)+ (p+ r) (again because 1 is true for V) and then
    = v+ u.
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