1. ## Vector Spaces

Hi. I have some homework that I am doing - we're doing the chapter on Vector Spaces. I've got most of them, not that I'm really understanding Vector Spaces very well, but he assigns a ridiculous amount of problems and a few of them have me stumped (below). Help would be greatly appreciated.

4.2
===
1) Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.

The set of all 2x2 nonsingular matrices with the standard operations.
I said true, it's a vector space. ????

2) True/False AND cite a reason.
a) To show that a set is not a vector space, it is sufficient to shwo that just one axiom is not satisfied.
Actually got this one. True - must satisfy all axioms by definition of a vector space.
b) The set of all first-degree polynomials with the standard operations is a vector space.
I'm guessing true, but unsure.
c) The set of all pairs of real numbers of the form (0,y), with the standard operations on R^2, is a vector space.
No clue. What do they mean by operations in R^2? Just adding and subtracting vectors? And try it for the 10 axioms?

3) Prove that in a given vector space V, the zero vector is unique.

Not that I could even do this anyways, but what do they mean by unique? There is only one <0,0> vector??? I don't get it.

4 Prove that in a given vector space V, the additive inverse of a vector is unique.

Just as clueless as the last one.

4.3
===
1) Determine is the subset of C(-infinity, infinity) is a subspace of C(-infinity, infinity) for the set of all functions such that f(0) = 1.

2) Prove that a nonempty set W is a subspace of a vector space V if and only if ax + by is an element of W for all scalars a and b and all vectors x and y in W.

3) Let A be a fixed m x n matrix. Prove that the set W = {x E R^3: Ax = [[1][2]]}

the [1,2] matrix there is a column - 2x1.

4) Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.

2. Originally Posted by kaylakutie
Hi. I have some homework that I am doing - we're doing the chapter on Vector Spaces. I've got most of them, not that I'm really understanding Vector Spaces very well, but he assigns a ridiculous amount of problems and a few of them have me stumped (below). Help would be greatly appreciated.

4.2
===
1) Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.

The set of all 2x2 nonsingular matrices with the standard operations.
I said true, it's a vector space. ????

2) True/False AND cite a reason.
a) To show that a set is not a vector space, it is sufficient to shwo that just one axiom is not satisfied.
Actually got this one. True - must satisfy all axioms by definition of a vector space.
b) The set of all first-degree polynomials with the standard operations is a vector space.
I'm guessing true, but unsure.
c) The set of all pairs of real numbers of the form (0,y), with the standard operations on R^2, is a vector space.
No clue. What do they mean by operations in R^2? Just adding and subtracting vectors? And try it for the 10 axioms?

3) Prove that in a given vector space V, the zero vector is unique.

Not that I could even do this anyways, but what do they mean by unique? There is only one <0,0> vector??? I don't get it.

4 Prove that in a given vector space V, the additive inverse of a vector is unique.

Just as clueless as the last one.

4.3
===
1) Determine is the subset of C(-infinity, infinity) is a subspace of C(-infinity, infinity) for the set of all functions such that f(0) = 1.

2) Prove that a nonempty set W is a subspace of a vector space V if and only if ax + by is an element of W for all scalars a and b and all vectors x and y in W.

3) Let A be a fixed m x n matrix. Prove that the set W = {x E R^3: Ax = [[1][2]]}

the [1,2] matrix there is a column - 2x1.

4) Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
You know that for every real No x ( to make matters simple) x+0 = x

Now how do you know that there is no other real No y such that x+y =x.

Since real No are infinite and one cannot have experience of all the real Nos

Also for every real No x there exists another No y such that x+y = 0

Again how can you be sure that this No is one and only one

3. Originally Posted by kaylakutie
Hi. I have some homework that I am doing - we're doing the chapter on Vector Spaces. I've got most of them, not that I'm really understanding Vector Spaces very well, but he assigns a ridiculous amount of problems and a few of them have me stumped (below). Help would be greatly appreciated.

4.2
===
1) Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.

The set of all 2x2 nonsingular matrices with the standard operations.
I said true, it's a vector space. ????
Yes, that's true. Of course you really should show that all axioms hold.

2) True/False AND cite a reason.
a) To show that a set is not a vector space, it is sufficient to shwo that just one axiom is not satisfied.
Actually got this one. True - must satisfy all axioms by definition of a vector space.
Yes!

b) The set of all first-degree polynomials with the standard operations is a vector space.
I'm guessing true, but unsure.
Be careful here. What, exactly, does "first-degree polynomial" mean? Is it "polynomials of degree 1 or less" or only "polynomials of degree 1"? In particular, is the 0 function f(x)= 0 for all x a "first degree" polynomial?

[quote][c) The set of all pairs of real numbers of the form (0,y), with the standard operations on R^2, is a vector space.
No clue. What do they mean by operations in R^2? Just adding and subtracting vectors? And try it for the 10 axioms?[quote]
The "standard operations on R^2" are (a,b)+ (c,d)= (a+c, b+ d) and r(a, b)= (ra, rb) for r any real number.

3) Prove that in a given vector space V, the zero vector is unique.

Not that I could even do this anyways, but what do they mean by unique? There is only one <0,0> vector??? I don't get it.
That's not the way a "zero-vector" is defined. The "zero-vector" is defined as the vector, v, such that for any other vector u, u+ v= u. If there were another vector, v1, which was also a "zero-vector" then u+ v1= u and so u= u+ v= u+ v1 Since one of the axioms of vector spaces is that every vector has an additive inverse (negative), add the inverse of u to both sides of u+ v= u+ v1.

4 Prove that in a given vector space V, the additive inverse of a vector is unique.

Just as clueless as the last one.
Suppose that the vector, u, has two different additive inverses, v1 and v2. By the definition of additive inverse, we would have u+v1= 0 and u+ v2= 0. Since we have just proven that the zero-vector is unique, we must have u+v1= u+ v2. Add v1 to both sides of that: v1+(u+v1)= v1+(u+ v2), (v1+ u)+ v1= (v1+u)+ v2, 0+ v1= 0+ v2, v1= v2.

Notice how I used the associative and commutative laws in that.

4.3
===
1) Determine is the subset of C(-infinity, infinity) is a subspace of C(-infinity, infinity) for the set of all functions such that f(0) = 1.
Is 0 function in that set?

2) Prove that a nonempty set W is a subspace of a vector space V if and only if ax + by is an element of W for all scalars a and b and all vectors x and y in W.
Since you are given that V is a subspace, you know that many of the properties, such as comutativity and associativity of addition, are true. To show that W is a subspace you really only need to show that
1) The set is non-empty: take a= b= 0: 0x+ 0v= 0 is in the set.
2) it is closed under addition: for vectors u and v in W, take a= b= 1 and x= u, y= v to show that u+ v is in W.
3) It is closed under scalar multiplication: for vector v in W and scalar r, take a= r, b= 0, x= v, y= 0 to show that ar is in W.

Since this is "if and only if" you also need to show the other way- suppose W is a subspace. Then if must be closed under scalar multiplication so for any scalars a and b and vectors x and y in W, both ax and by are in W. But a subspace is also closed under addtion so ax+ by is in W.

[quote]3) Let A be a fixed m x n matrix. Prove that the set W = {x E R^3: Ax = [[1][2]]}

the [1,2] matrix there is a column - 2x1.
That can't be write. If W is in $\displaystyle R^3$, A cannot be "m x n". And prove that the set W is what?

4)0 Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Take V= $\displaystyle R^2$, the space of pairs of numbers, (x, y) with (x, y)+ (u, v)= (x+u, y+ v) and r(x, y)= (rx, ry).

Show that U= {(x, 0)} and W= {(0, y)} are subspaces. What is the union of U and W?

4. Wow, thanks a ton!

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# Determine whether the set of all 2×2 singular matrices with the standard operations is a vector space. If not, identify at least one of the ten vector space axioms that fails.

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