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Vectors & Linear Transformations

59 cards · 🇺🇸 English

Vector operations and linear transformations/matrices.

linear-algebravectorstransformations

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Vector space

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  1. 1Vector space
    A set whose elements (vectors) can be added together and multiplied by scalars, satisfying eight axioms including associativity, commutativity, identity elements, and distributivity.
  2. 2Two fundamental operations in a vector space
    Vector addition (assigning v + w to any two vectors) and scalar multiplication (assigning av to a scalar a and vector v).
  3. 3Zero vector
    An element 0 ∈ V such that v + 0 = v for all vectors v in the vector space. It's the identity element of vector addition.
  4. 4Additive inverse of a vector v
    An element −v such that v + (−v) = 0. Every vector in a vector space must have an additive inverse.
  5. 5Scalar in the context of vector spaces
    An element of the field (typically real or complex numbers) by which vectors are multiplied in scalar multiplication.
  6. 6Linear combination
    An expression of the form a₁v₁ + a₂v₂ + ... + aₙvₙ where aᵢ are scalars and vᵢ are vectors.
  7. 7Vectors to be linearly independent
    No element of the set can be written as a linear combination of the others; equivalently, a linear combination equals zero if and only if all coefficients are zero.
  8. 8Linear subspace
    A non-empty subset of a vector space that is closed under vector addition and scalar multiplication (the sum and scalar multiples of elements remain in the subset).
  9. 9Span of a set of vectors
    The smallest linear subspace containing the set; equivalently, the set of all linear combinations of elements in that set.
  10. 10Basis of a vector space
    A subset whose elements are linearly independent and span the entire vector space.
  11. 11Dimension of a vector space
    The cardinality (number of elements) of any basis; all bases of a vector space have the same cardinality.
  12. 12Coordinate vectors
    The n-tuple of scalars (coefficients) that uniquely represent a vector as a linear combination of basis vectors.
  13. 13Linear map (linear transformation)
    A function f between vector spaces that preserves sums and scalar multiplication: f(u + v) = f(u) + f(v) and f(av) = af(v).
  14. 14What property does every linear map have regarding the origin?
    A linear map always maps the origin of the domain to the origin of the codomain: f(0) = 0.
  15. 15Isomorphism between vector spaces
    A linear map that is bijective (both one-to-one and onto), meaning it has an inverse that is also linear.
  16. 16Linear endomorphism
    A linear map from a vector space to itself: f : V → V.
  17. 17Kernel (null space) of a linear map f : V → W
    The set of all vectors in V that map to zero in W: ker(f) = {v ∈ V : f(v) = 0}. It's a subspace of V.
  18. 18Image (range) of a linear map f : V → W
    The set of all vectors in W that are the image of some vector in V: im(f) = {f(v) : v ∈ V}. It's a subspace of W.
  19. 19State the rank-nullity theorem.
    For a linear map f : V → W where V is finite-dimensional: dim(V) = dim(ker(f)) + dim(im(f)), or dimension of domain = nullity + rank.
  20. 20Rank of a linear transformation
    The dimension of the image (range) of the transformation, denoted rank(f) or dim(im(f)).
  21. 21Nullity of a linear transformation
    The dimension of the kernel (null space) of the transformation, denoted nullity(f) or dim(ker(f)).
  22. 22How are linear maps represented using matrices?
    Once bases are chosen for domain and codomain, a linear map is uniquely represented by a matrix where columns contain the images of the basis vectors.
  23. 23How does matrix multiplication relate to linear maps?
    The composition of linear maps corresponds to matrix multiplication; if f and g are linear maps, the matrix of g∘f is the product of their matrices.
  24. 24What does the determinant of a matrix tell about the corresponding linear map?
    Whether the map is an isomorphism (bijective); the map is an isomorphism if and only if the determinant is non-zero.
  25. 25Eigenvector of a linear transformation f : V → V
    A non-zero vector v such that f(v) = λv for some scalar λ (the eigenvector is scaled by the transformation).
  26. 26Eigenvalue
    The scalar λ associated with an eigenvector v, satisfying f(v) = λv for a linear endomorphism f.
  27. 27How are eigenvalues computed for a matrix A?
    By finding the roots of the characteristic polynomial det(A - λI) = 0, where I is the identity matrix.
  28. 28Eigenspace
    The set of all eigenvectors corresponding to a particular eigenvalue, together with the zero vector. It forms a vector subspace.
  29. 29Dual vector space V*
    The space of all linear functionals (linear maps) from V to the field F, denoted V* = Hom(V, F).
  30. 30Linear functional
    A linear map from a vector space to its underlying field of scalars, f : V → F.

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