Math 340 UW Madison Midterm 2 Flashcards

Vector Space

• Defined as a set V of elements closed under vector addition and scalar multiplication.

Vector Space Properties

• Vector addition:
  • Commutative: u + v = v + u
  • Associative: u + (v + w) = (u + v) + w
  • Identity: Exists a zero element such that u + 0 = 0 + u = u
  • Inverses: For each u, there exists -u such that u + (-u) = 0
• Scalar multiplication:
  • Distributive over vector addition: c * (u + v) = c * u + c * v
  • Distributive over scalar addition: (c + d) * u = c * u + d * u
  • Associative: c * (d * u) = (cd) * u
  • Identity: 1 * u = u for any u in V

Vector Subspace

• A nonempty subset W of V is a vector subspace of V if it is a vector space itself under the operations of V.

Linear Combination

• A vector v is a linear combination of vectors v1, ..., vk if v = a1v1 + a2v2 + ... + ak*vk, where a1, a2, ..., ak are scalars.

Solution Space / Null Space

• Given a matrix A, the null space W comprises all solutions to the equation Ax = 0.

Span

• The span of a set S of vectors in a space V consists of all vectors formed by linear combinations of vectors in S.

Spanning Set

• A set S is a spanning set of V if span(S) = V, indicating that V can be fully expressed as a linear combination of vectors in S.

Linear Independence

• A set of vectors {v1, v2, ...} is linearly independent if the only solution to a1v1 + a2v2 + ... = 0 is a1 = a2 = ... = 0.
• Determinant approach: A set S is linearly independent if the determinant of the matrix formed by its vectors (as columns) is non-zero.
• Preceding vectors method: A nonzero vector vj is linearly dependent if it can be expressed as a combination of preceding vectors.

Basis

• A basis for a vector space V consists of a set of vectors S that span V and are linearly independent.

Dimension

• The dimension of a vector space is defined as the number of vectors in a basis for that space.

Ordered Basis

• A basis where the sequence of vectors is fixed, providing a unique representation for each vector in the space.

Coordinate Vector

• The coordinate vector of a vector v with respect to an ordered basis S is represented as [v]S = (a1, a2, ..., an), indicating v can be expressed as v = a1v1 + a2v2 + ... + an*vn.

One-to-One (Injective)

• A function f is injective if f(x) = f(y) implies x = y.

Onto (Surjective)

• A function f from set A to B is surjective if for every b in B, there exists at least one x in A such that f(x) = b.

Bijective

• A function is bijective if it is both injective and surjective.

Isomorphism

• A bijective function f between vector spaces V and W is an isomorphism if it preserves addition and scalar multiplication. Equal dimensions are required for vector spaces to be isomorphic.

Column/Row Space

• For an mxn matrix A, the column space is the span of the columns, while the row space is the span of the rows, forming subspaces of R^m and R^n, respectively.

Column/Row Rank

• The rank of a matrix A, determining the dimension of its column or row space, is equivalent to the number of nonzero rows in its row echelon form.

Nullity

• Nullity is the dimension of a matrix's null space. For an mxn matrix, Rank + Nullity = n.

Linear Transformation

• A function L: V → W is a linear transformation if it preserves both vector addition and scalar multiplication. Every isomorphism is a linear transformation.

Kernel

• The kernel of a linear transformation L consists of all vectors v in V that satisfy L(v) = 0. The kernel is a subspace of V.

Image/Range

• The image (or range) of a linear transformation L: V → W consists of all vectors in W that can be expressed as L(v) for some v in V.

Nonsingular Matrix Properties

• A nonsingular matrix A has the following properties:
  • The equation Ax = 0 has only the trivial solution.
  • It is row/column equivalent to the identity matrix I_n.
  • The linear system Ax + b has a unique solution for every b.
  • A can be expressed as a product of elementary matrices.
  • Determinant det(A) is non-zero.
  • Rank of A is equal to n and nullity is zero.
  • Rows and columns of A are linearly independent, considered as vectors in R^n.
  • The associated linear transformation L: R^n → R^n defined by L(v) = A(v) is both one-to-one and onto.