Linear subspaces

• Goal: study properties of the model space (, the Euclidean space)
  • Understand the definition of vector spaces
• is the set of all real numbers
  • we can add two real numbers and if we add or subtract them, they are still real numbers
    • This means real numbers are closed under addition
  • Same thing for , real numbers are closed under scalar multiplication
• is the set of all ordered pairs:
  • We can think of as a point
    • We could think of this as an absolute position on the space
  • Or, we can think of as a vector.
    • Instead, we think of it as a translation
    • It has a magnitude and direction
• Points cannot be added, but vectors can
  • Under the definitions for vector addition and subtraction, we can say that is closed under vector addition
• • Vector addition and subtraction is defined in the exact same way in
  • is closed under vector addition and subtraction for any
  • This property is not unique for a finite , it works for when is infinite as well
    • say
    • is still closed under addition
    • is the space of all “sequences” of real numbers
• , as well as are closed under scalar multiplication as well
• is also closed under scalar multiplication
  • say , multiplying it by some would be
• There are subsets of ,
  • For example:
    • This would be the -plane
  • The set would define a line
• Other subsets include:
  • This is the unit (n-1) sphere in :
    • In , this would be a two points at 1 and -1.
    • In , this would be the 2d unit circle that we know:
      • We denote this as
      • This is a 1 dimensional object, even though it seems like 2 dimensions. This is because at one point, you can only back and forward
      • If you take the tangent to the circle, you get a line
    • In , we get an actual sphere
      • This is a 2 dimensional object, because when you zoom in on the surface, it is flat like a plane
      • Think about a tangent plane to the sphere
      • 

Note: is NOT a subset of or , because they contain tuples of different lengths. As a result, no real number is in the set of tuples, and no 2-tuple is in the set of 2 tuples, and so on so forth. Definition: Subspaces. A subspace of is a subset of with the property:

1. if , then
2. if then

• Let be a subspace
  • We know that
  • , is a subspace of itself
  • These two examples are trivial subspaces of

Non-example: Circle: . If we add two elements from this, , we get , which is not in

Midterm review

What is the smallest subspace that contains subspaces ? it is called . It is the smallest if is a subspace containing them also contains . . Pick . , where . , where For any , . Expand this as We know that we can group them like this to see that the sums of the elements are contained by . We know that is the smallest subspace containing and . Say that we have some is a subspace with and . Pick, . Since Therefore,

Say that and Pick some . Pick Then Therefore,