Lecture 3:

  • Basis

    • In a Vector Space V, there exist vectors called basis that satisfy the following conditions:
      • Definition: “The vectors can generate any vector in V through linear combinations” and “The vectors are linearly independent”.
    • What is the difference between being linearly independent and being a basis..? (blu3mo)
      • When we say a set of linearly independent vectors, is the number of vectors not specified? (blu3mo)
      • It seems that we can define it as “a set of vectors that is linearly independent and has the same number as the dimension of the space”.
        • Oh, but is the concept of dimension not defined yet?
        • Maybe the dimension is defined by the definition of a basis? (blu3mo)
    • When expressing a vector using a certain basis, the representation is uniquely determined.
      • Well, that’s obvious (blu3mo)
      • For an orthonormal basis, there is no other representation of (5,3) except for 5(1,0) + 3(0,1).
      • For a skew-orthogonal basis, there is no other representation of (5,3) except for 2(1,0) + 3(1,1).
    • It is useful to have various bases when considering Diagonalization.
    • In physics, I don’t think I’ve ever considered anything other than an orthonormal basis (blu3mo).
      • I have considered a skew-orthogonal basis for the same unit, but I wonder if there are any advantages to considering a skew-orthogonal basis for vectors like velocity and time.
        • I think I used skew-orthogonal bases in relativity theory (takker).
        • Sounds interesting (blu3mo)(blu3mo)(blu3mo).
      • It is also necessary to discuss curved spaces (takker).
        • Relativity theory is famous, but it is also used when considering the deformation of surfaces in continuum mechanics.
  • Extension of Basis

  • Dimension

    • I’ve never thought about the definition of dimension (blu3mo).
    • As a premise, the number of bases of the same vector space is constant.
      • Therefore, the number is defined as the dimension.
      • It’s casually mentioned as a premise, but it actually needs to be proven (takker).
        • However, it’s not a difficult proof for finite-dimensional vector spaces.
          • It was proven in the class (blu3mo).
          • That’s good to hear (takker).
        • It seems to be a problem for infinite-dimensional vectors.
          • How can we prove it?
          • Can we simply show that a bijection can be constructed between two bases for the same vector space?
    • Theorem
      • They all seem obvious intuitively.
      • Theorem: If there are n vectors () in a vector V, they are linearly dependent.
      • Theorem: If we consider a subspace W of vector V, .

Lecture 2:

  • Subspace = Subvector Space
    • When we have a vector space V, a subset that itself becomes a vector space.
      • For example,
        • When considering a subspace of natural numbers, if there is only 3, it does not satisfy one of the axioms, so it does not become a subspace (blu3mo).
          • It becomes a subspace if both 3 and -3 are included (blu3mo).
        • It is not a subspace if it does not contain 0 (zero element).
      • Definition/Requirements of a subspace
          1. It contains 0.
          • Is this necessary? (takker)
            • If we exclude the empty set from being a subspace, can it be replaced with “not an empty set”?
          1. When there are two elements, their sum is also included in the set.
          • (Without this, addition does not hold)
          1. When there is an element, any scalar multiple of it is also included in the set.
          • (Without this, scalar multiplication does not hold)
        • As long as we consider these three, we can determine if it is a subspace (blu3mo)(blu3mo).
          • Since it is a subset of a vector space, we can skip checking things like the associative law (blu3mo).
          • The condition for the inverse element can also be included by considering -1 times in 3) (blu3mo).
    • The subspaces of are only the following three.- > , a line passing through the origin,
  • Each of them is 0-dimensional, 1-dimensional, and 2-dimensional (blu3mo)(blu3mo)
  • Most cases fall into either 2) or 3) (blu3mo)
  • It might be interesting to also examine subspaces of (takker)
  • Linear Independence
    • Linear Combination: When is an element of the vector space V, it can be expressed in the form
      • a is a scalar, right? (blu3mo)
      • Yes, it is a scalar (takker)
    • Linear Independence: ” implies
      • When there are two 2D vectors pointing in different directions, we can say that “if the linear combination is zero, then the scalar coefficients of each vector are also zero”
      • Pay attention to the logic (blu3mo)
      • Well, in this case, the reverse is also true, but it’s important to pay attention to the direction (takker)
    • Linear Dependence: not linearly independent
    • Hypothesis: For n-dimensional vectors, it seems that n different vectors pointing in separate directions are linearly independent (blu3mo)
      • Seems like it (blu3mo)
    • As a theorem, there is the following:
    • Theorem: ” are linearly independent” is equivalent to ” are not on the same line”

      • (Here, vector)
      • This can be proven by manipulating the equations
      • It seems that the same can be said for (placing n vectors)
      • Regarding the number n:
        • In the vector space , n+1 or more vectors cannot be linearly independent
        • This is because with n linearly independent vectors, any vector can be expressed as a linear combination of them