Questions and Answers
What is a subspace?
A subset H of some vector space V which includes the zero vector of V, is closed under vector addition, and closed under multiplication by scalars.
Is Span a subspace if the vectors are in the vector space?
True
How do you find a spanning set?
Solve Ax=0.
How do you check if Span {v1, v2, v3} is in R3?
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What is Nul A?
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How do you find Nul A?
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If Nul A = 0, what does it imply?
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How can you find the spanning set of Nul A?
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What is the Kernel T?
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What is Col A?
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Are both Nul A and Col A subspaces?
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Does Col A span Rn?
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How can you determine if u is in Col A?
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What is the Range of T?
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What does linearly independent mean?
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What does linearly dependent mean?
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What is a basis?
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What is a standard basis?
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How do you find a basis in general?
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What is the Basis for Nul A?
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What is the Basis for Col A?
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What is the Spanning Set Theorem?
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What does the Unique Representation Theorem state?
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What are B-coordinates of x?
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What is coordinate mapping?
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What is change of basis?
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Study Notes
Subspace
- A subset H of vector space V qualifies as a subspace if it includes the zero vector, is closed under vector addition, and is closed under scalar multiplication.
Span
- Span is a subspace formed from vectors within a vector space, requiring the zero vector to be included in the Span.
Finding a Spanning Set
- To find a spanning set, solve the equation Ax=0.
Checking Span in R3
- To verify if Span{v1, v2, v3} exists in R3, ensure there is a pivot in every row of the matrix.
Nul A
- Nul A is the set of all solutions to the equation Ax=0, which always includes the trivial solution (the zero vector).
Finding Nul A
- Determine Nul A by solving Ax=0.
Conditions for Nul A
- If Nul A equals {0}, the mapping is one-to-one, indicating that the columns of A are linearly independent.
Spanning Set of Nul A
- The spanning set for Nul A is obtained by solving Ax=0, with the number of vectors equating to the number of free variables.
Kernel T
- The kernel of a transformation T is synonymous with Nul A.
Col A
- Col A consists of all linear combinations of the columns of matrix A, equivalent to Span{a1, ..., an}.
Properties of Nul A and Col A
- Both Nul A and Col A are subspaces and thus contain the zero vector.
Col A Spanning Rn
- Col A spans Rn if there is a pivot in every row of the matrix.
Checking Membership in Col A
- To determine if a vector u belongs to Col A, perform the operation [A u]; if the last column is not a pivot, then u is in Col A.
Range of T
- The range of transformation T is represented by Col A.
Linear Independence
- Vectors are linearly independent if they have solely the trivial solution and lack a pivot in every column.
Linear Dependence
- Vectors are linearly dependent if they possess a nontrivial solution that indicates a dependency relationship among them, with pivots in every column.
Basis
- A basis for a subspace consists of linearly independent vectors whose span equals the subspace, with columns of an invertible matrix forming a basis.
Standard Basis
- The standard basis is represented as E = {e1,..., en} where e_i are unit vectors.
Finding a Basis
- To find a basis, every row and column of the matrix must contain a pivot, ensuring linear independence and spanning Rn.
Basis for Nul A
- A basis for Nul A is obtained by solving Ax=0 using reduced row echelon form (rref), yielding a linearly independent spanning set.
Basis for Col A
- The basis for Col A includes the pivot columns of the original matrix A, confirming their linear independence.
Spanning Set Theorem
- A set S={v1,...,vp} spans H if removing any vector vk that is a linear combination of others still spans H. Additionally, if H≠{0}, some subset of S forms a basis for H.
Unique Representation Theorem
- Each vector x in a vector space V can be expressed uniquely as a linear combination of the basis B coefficients: x = c1b1 + ... + cnbn.
B-coordinates of x
- B-coordinates of x refer to its coefficients/scalars c1...cn in the linear combination.
Coordinate Mapping
- Coordinate mapping connects each vector x in V to its coordinate vector [x]B, satisfying the relationship x = PB[x]B, with properties of being one-to-one and onto.
Change of Basis
- When transitioning between two bases B and C, a matrix P exists as a transformation.
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Prepare for your MAS3114 exam with these flashcards focused on key concepts in vector spaces. Each term includes a definition that clarifies essential properties and methods related to subspaces and spanning sets. Test your understanding and improve your grasp of the material before the exam.