MAS3114 Exam 2 Flashcards

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?

Look for a pivot in every row. Signup and view all the answers

What is Nul A?

The set of all solutions to Ax=0. Signup and view all the answers

How do you find Nul A?

Solve Ax=0. Signup and view all the answers

If Nul A = 0, what does it imply?

One-to-one mapping with ONLY the trivial solution; the columns of A are linearly independent. Signup and view all the answers

How can you find the spanning set of Nul A?

Solve Ax=0; the number of vectors in the spanning set equals the number of free variables. Signup and view all the answers

What is the Kernel T?

Nul A. Signup and view all the answers

What is Col A?

The set of all linear combinations of the columns of A; it is equal to Span{a1,..., an}. Signup and view all the answers

Are both Nul A and Col A subspaces?

True Signup and view all the answers

Does Col A span Rn?

Yes, if there is a pivot in every row. Signup and view all the answers

How can you determine if u is in Col A?

Solve [A u]; if the last column is NOT a pivot, then yes. Signup and view all the answers

What is the Range of T?

Col A. Signup and view all the answers

What does linearly independent mean?

False Signup and view all the answers

What does linearly dependent mean?

False Signup and view all the answers

What is a basis?

A set of vectors is a basis for a subspace if they are linearly independent and H = Span{b1,..., bp}. Signup and view all the answers

What is a standard basis?

The basis E = {e1,..., en}. Signup and view all the answers

How do you find a basis in general?

Every row and column must be a pivot. Signup and view all the answers

What is the Basis for Nul A?

Solve Ax=0 using rref, which produces a linearly independent spanning set. Signup and view all the answers

What is the Basis for Col A?

It consists of the pivot columns of A (the original matrix). Signup and view all the answers

What is the Spanning Set Theorem?

If H doesn't equal {0}, some subset of S is a basis for H. Signup and view all the answers

What does the Unique Representation Theorem state?

For each x in V, there exists a unique set of scalars c1,..., cn such that x = c1b1 +...+ cnbn. Signup and view all the answers

What are B-coordinates of x?

They are the coefficients/scalars c1...cn. Signup and view all the answers

What is coordinate mapping?

It maps each x in V to its coordinate vector [x]B; the mapping is one-to-one and onto. Signup and view all the answers

What is change of basis?

When two bases B and C exist, there exists a transformation P such that C transforms to B. Signup and view all the answers

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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Description

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.