Questions and Answers
What is a Vector Space?
What are the properties of vector addition in a Vector Space?
u + v = v + u, u + (v + w) = (u + v) + w, there exists 0 such that u + 0 = 0 + u = u, for each u there exists -u such that u + -u = 0.
What are the properties of scalar multiplication in a Vector Space?
c * (u + v) = c * u + c * v, (c + d) * u = c * u + d * u, c * (d * u) = (cd) * u, 1 * u = u for any u.
If W is a nonempty subset of V and is a vector space with respect to the operative rules of V, W is a ________ of V.
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A vector v is a ________ of v1...vk if v=a1v1+a2v2...+akvk.
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If A is a mxn matrix and W is the set of all solutions to Ax=0, then W is the ______________ of A.
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If S is a set of vectors in a space V, then the set of all vectors of V that are a linear combination of the vectors of S is ________.
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If the set of vectors S is in vector space V, then span S is a _________ of V.
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If span S = V then S is a _________ of V.
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The vectors v1, v2, etc. are ____________ if there exists no real numbers a1, a2, etc. (besides 0) such that a1v1 + a2v2 + ... + anvn = 0.
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If S is a set of vectors, let A be the matrix whose columns are the elements of S. S is then __________ iff det A DNE 0.
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When are nonzero vectors v1, v2, ...vj linearly dependent?
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The set of vectors S in vector space V are said to form a ______ for V iff S spans V and S is linearly independent.
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_________ of a nonzero vector space is the number of vectors in a basis for V.
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Basis in which the order of the vectors is fixed is called __________.
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The ______________ of v with respect to the ordered basis S is [v]S (a1, a2, ..., an) such that v = a1v1 + a2v2 + ... + anvn.
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If f(x) = f(y) implies x = y, then f is called one-to-one.
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A transformation is called onto if for every b in B there exists at least one x in A such that f(x) = b.
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If a function is both injective and surjective, it is called bijective.
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What is an Isomorphism?
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If A is an mxn matrix, then the columns/rows of A span a subspace of R^m/n called the / of A.
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The number of nonzero rows in the row/column space of A is the _______ of A.
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For a matrix A, the dimension of the null space of A is the _________ of A.
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What is a Linear Transformation?
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The ________ of a linear transformation L is the subset of V consisting of all elements v of V such that L(v) = 0.
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If L: V --> W is a linear transformation, then the _______ of V under L consists of all vectors in W that are images under L of vectors in V.
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What are the key properties of a nonsingular matrix A?
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Study Notes
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.
Studying That Suits You
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Description
Test your understanding of key concepts in vector spaces with these flashcards. Each card covers essential properties of vector addition and scalar multiplication. Perfect for reviewing before the midterm exam in Math 340 at UW Madison.
