MAS3114 Exam 2 Flashcards

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.