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Kuma
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I have a question that asks me to show that something is a linear space. but what is a linear space exactly and how does it differ from vector space or euclidian space?
A subspace is by definition a vector space, so if you just show that 0 is in L and that L is closed under linear combinations, then the theorem ensures that L is a vector space. And in this case, you said that your L was defined as a set of "vectors". To me that can only mean that L is defined as a subset of a vector space, but perhaps you didn't mean to imply that by using the word "vectors".Kuma said:I have learned that theorem, but how would it apply here? I'm asked to prove that L is a vector space itself, so shouldn't I use the definition of a vector/Linear space itself to show that it indeed is a vector space? ie proving the axioms for all elements in L?
However, you still need to prove that subset is non-empty and typically the simplest way to do that is to prove that 0 is in the set.Fredrik said:Edit: The requirement that 0 is in U is unnecessary, because the requirement that ax+by is in U for all a,b in ℝ and x,y in U implies that 0=0x+0y is in U.
Ah, good point. Thanks. All vector spaces have a member denoted by 0, so no vector space is empty. That makes my (a) not equivalent to (b) and (c), since U=∅ satisfies (b) and (c) but not (a). So I need to add something like U≠∅ to (b) and (c)...but it looks nicer to require that 0 is in U.HallsofIvy said:However, you still need to prove that subset is non-empty and typically the simplest way to do that is to prove that 0 is in the set.
A linear space, also known as a vector space, is a mathematical concept that refers to a collection of objects called vectors that can be added together and multiplied by a scalar to produce another vector within the same space. This space follows a set of axioms and properties, allowing for mathematical operations to be performed on the vectors.
Some common examples of linear spaces include the Cartesian plane, where points can be added and multiplied by a scalar to produce new points, and the space of real numbers, where numbers can be added and multiplied to produce new numbers. Other examples include function spaces, such as the space of all polynomials of a certain degree, and matrix spaces, where matrices can be added and multiplied by scalars to produce new matrices.
A vector is an element within a linear space. In other words, a vector is a specific object that belongs to a larger linear space. The linear space itself is a collection of all possible vectors that can be formed by following the defined axioms and properties.
The fundamental properties of a linear space include closure under vector addition and scalar multiplication, associativity and commutativity of addition, distributivity of scalar multiplication, and the existence of an additive identity element (usually represented by the zero vector). These properties allow for the manipulation and combination of vectors within the space.
While both linear spaces and Euclidean spaces involve the combination and manipulation of vectors, a Euclidean space specifically refers to a geometric space with a defined set of axes and a specific metric, such as the Cartesian plane. A linear space, on the other hand, is an abstract mathematical concept that can be applied to various types of spaces, including Euclidean spaces, but is not limited to them.