Vector space, linear transformations & subspaces

In summary: You're right that you should probably show that, but it's a pretty minor detail. In summary, the conversation discusses the problem of proving that the subset W, consisting of vectors in a vector space V that are mapped by two linear transformations L and M to the same vector, is a subspace of V. The participants explore the properties of subspaces and linear transformations, including closure under addition and scalar multiplication, and the existence of a zero vector. The conversation reveals some confusion around the definitions and requirements for proving these properties, leading to a lack of clarity in the attempted solution.
  • #1
ilyas.h
60
0

Homework Statement



Let V be a vector space over a field F and let L and M be two linear transformations from V to V.

Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .

The Attempt at a Solution



I presume it's a simple question, but it's one of those where you just don't know where to start. This is my attempt;

Subspace:
-0 vector exists
-closed under addition
-closed under scalar multiplicationLinear transformation:
-x,y in V: L(x+y) = L(x) + L(y)
-a in F, x in V: L(ax) = aL(x)
Proving 0 vector exists:

L & M are linear transformations so:

L(x+y) = L(x) + L(y)

but since L(x) = M(x):

L(x+y) = M(x) + M(y) = M(x+y)

∴ L(x+y) + (-M(x+y)) = 0_v

Also:

L(ax) = aL(x)

but since L(x) = M(x):

L(ax) = aM(x) = M(ax)

∴ L(ax) + (-M(ax)) = 0_v

Therefore, 0 vector exists in W.That's all I've got. Don't know how to prove closed under addition and closed under scalar multiplication.
 
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  • #2
to show that it is closed you need to show that if x, y ∈ W and a ∈ F then x + y ∈ W and ax ∈W also but you've already done this by showing L(x + y) = M(x + y) etc above.
 
  • #3
fourier jr said:
to show that it is closed you need to show that if x, y ∈ W and a ∈ F then x + y ∈ W and ax ∈W also but you've already done this by showing L(x + y) = M(x + y) etc above

so essentially I've done the problem but in an "all in one" fashion?
 
  • #4
I don't think that shows that 0 is in W though, since you need to have L(0) = M(0) for that to be true. I would start L(0) = L (x - x) = ...

& I would also skip the lines that say ∴ L(x+y) + (-M(x+y)) = 0_v because the previous lines are enough
 
  • #5
ilyas.h said:
so essentially I've done the problem but in an "all in one" fashion?
No. It seems like you don't really have a clear idea of what you need to show, which makes writing a proof more difficult because you don't know where to start and where you're trying to get to.

To show that W is closed under vector addition, for example, you need to show that if x and y are in W, then x+y is in W. "x and y are in W" is your starting point, so you should start with the statement "Let x and y be in W." Note that in your attempts, you never said anything about what x and y are. For all we know from what you wrote, they are random vectors in V, so your assertion that L(x) = M(x) isn't justified. On the other hand, if you explicitly say x is in W, then by the definition of W, you know that L(x) = M(x).

"x+y is in W" is where you're trying to get to. This means you need to show that L(x+y) = M(x+y). When you get to that line, you can then conclude that x+y is in W based on the definition of W.

To prove that 0 is in W, what do you need to show? To prove W is closed under scalar multiplication, what do you need to show? Get what those mean clear in your mind, and the proofs will probably follow pretty easily, and you won't be wondering if you succeeded in showing what you were supposed to.
 
  • #6
fourier jr said:
I don't think that shows that 0 is in W though, since you need to have L(0) = M(0) for that to be true. I would start L(0) = L (x - x) = ...

& I would also skip the lines that say ∴ L(x+y) + (-M(x+y)) = 0_v because the previous lines are enough

But, T(0) = 0 for any linear transformation T, so L(0)=M(0) (both =0).
 

What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and two operations, addition and scalar multiplication. These operations follow specific rules, such as closure and associativity, and the vectors must satisfy certain properties, such as having a zero vector and being closed under scalar multiplication.

What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the vector space structure. This means that the transformation must preserve addition and scalar multiplication. It is often represented by a matrix, and can be thought of as stretching, rotating, or reflecting a vector space.

What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it must contain a zero vector, be closed under addition and scalar multiplication, and be closed under linear combinations. Subspaces can be thought of as smaller vector spaces within a larger one.

How do I determine if a set of vectors forms a basis for a vector space?

A basis for a vector space is a set of linearly independent vectors that span the entire vector space. To determine if a set of vectors forms a basis, you can perform row reduction on a matrix with these vectors as columns. If the matrix has a pivot in every row, then the vectors form a basis for the vector space.

What is the difference between a row space and a column space?

A row space is the span of the rows of a matrix, while a column space is the span of the columns. This means that the row space is the set of all linear combinations of the rows, while the column space is the set of all linear combinations of the columns. In general, the row space and column space of a matrix are different, but they have the same dimension and contain the same vectors if the matrix is square.

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