ASU MAT 343 Exam 2 Flashcards

Questions and Answers

What is a set V?

A collection of objects (correct)

Only numbers

A single object

An empty set

What are the two operations objects in the set V can be manipulated with?

o-plus and o-dot (correct)

exponentiation and root

multiplication and division

addition and subtraction

What does 'o-plus' act like?

addition

What does 'o-dot' act like?

multiplication Signup and view all the answers

'o-plus' defines...

Signup and view all the answers

'o-dot' defines...

Signup and view all the answers

What do the closure conditions guarantee?

That it exists in V and is a linear combination Signup and view all the answers

What is a vector space?

A nonempty set of V of objects that is closed under 'o-plus' and 'o-dot' Signup and view all the answers

What is Axiom 1 (A1)?

Commutativity of 'o-plus' Signup and view all the answers

What is Axiom 2 (A2)?

Associativity of 'o-plus' Signup and view all the answers

What is Axiom 3 (A3)?

Existence of a unique 0 vector which is neutral for 'o-plus' Signup and view all the answers

What is Axiom 4 (A4)?

A unique additive inverse -x exists for every x Signup and view all the answers

What is Axiom 5 (A5)?

Associativity of 'o-dot' Signup and view all the answers

What is Axiom 6 (A6)?

Distributivity of 'o-dot' with respect to the sum of scalars Signup and view all the answers

What is Axiom 7 (A7)?

Distributivity of multiplication with respect to 'o-plus' Signup and view all the answers

What is Axiom 8 (A8)?

1 is neutral for 'o-dot' Signup and view all the answers

How is Axiom 8 (A8) described?

Multiplicative identity Signup and view all the answers

If Axiom 3 (A3) fails, which axiom will automatically fail?

Axiom 4 because there is no zero vector Signup and view all the answers

If V is closed under 'o-plus' and 'o-dot' and A1-A8 are satisfied, what does that mean?

V is a vector space and its objects are vectors. Signup and view all the answers

What is the definition of a subset?

A Subset S of a vector space V is a subspace of V if it is closed under the two vector space operations 'o-plus' and 'o-dot' Signup and view all the answers

What analogy can we use to describe S?

'S can be viewed as a slice of V' Signup and view all the answers

What must S contain?

The zero vector Signup and view all the answers

Since S inherits the vector space structure from V, what do we know about the axioms?

The axioms are automatically satisfied Signup and view all the answers

V and the {0v} (0 vector) are subspaces of V and all other subspaces are...

proper subspaces Signup and view all the answers

What is the span of a set of vectors?

The set of all linear combinations of vectors {a1.....an} in V Signup and view all the answers

Is the span of a set of vectors closed?

True Signup and view all the answers

A span of a set of vectors is the ____ subspace of V containing the vectors in the set.

smallest Signup and view all the answers

If the span of {a1......an} = V, then:

{a1.....an} is called a spanning set of V Signup and view all the answers

{a1.....an} is called a spanning set of V if and only if:

Any vector b in V can be written as a linear combination of a1....an. Signup and view all the answers

What is the nullspace of a matrix?

The set of all vector solutions of the homogeneous system Ax = 0 Signup and view all the answers

Vectors are linearly independent if and only if:

The equation x1a1 + ...... + xnan = 0 vector only has the trivial solution Signup and view all the answers

Any set of vectors containing the zero vector is:

Linearly dependent Signup and view all the answers

A single non-zero vector is:

Linearly independent Signup and view all the answers

Two vectors a1 & a2 are linearly dependent if and only if:

One is a multiple of the other Signup and view all the answers

In a set of linearly dependent vectors at least one of them can:

Be written as a linear combination of the others Signup and view all the answers

For linearly dependent vectors, then the span can be seen as:

span{a1,a2,a3} = span{a1,a2} = span{a1,a3} = span{a2,a3} Signup and view all the answers

Adding vectors to a set of linearly dependent vectors...

Still yields a set of linearly dependent vectors Signup and view all the answers

A system that is underdetermined & homogeneous...

Will always have infinitely many solutions and therefore will not be linearly independent Signup and view all the answers

A spanning set of linearly independent vectors forms a...

Basis Signup and view all the answers

A basis requires:

Spanning set & linear independence Signup and view all the answers

A basis is a...

Minimal spanning set - contains the least number of vectors among all spanning sets Signup and view all the answers

If B = {a1.....an} is a basis of V, then...

Signup and view all the answers

What is the dimension of a vector space?

All bases of a vector space V have the same number of vectors n. That number (n) is the dimension written as dim(n). It can be finite or infinite. Signup and view all the answers

The standard basis is similar to the...

Identity matrix Signup and view all the answers

For R3, the standard basis is:

Signup and view all the answers

What is the spanning set (vectors)?

Any vector x can be written as a linear combination of the vectors in B. Signup and view all the answers

What is the linearly independent set?

The only solution is x1 = x2 = x3 = 0 Signup and view all the answers

What is the general rule for the dimension for Rn?

Signup and view all the answers

What is the standard basis for R2x2?

Signup and view all the answers

What is the spanning set (matrices)?

Any matrix can be written as a linear combination of the matrices in B. Signup and view all the answers

What is the standard dimension rule for Rmxn?

Signup and view all the answers

If dim(V) = n then...

Any set of n linearly independent vectors is a basis for V. Signup and view all the answers

What is the dimension of a subspace?

The dimension of the whole space minus the number of linearly independent constraints Signup and view all the answers

When using the standard basis, the representation vector will be...

The vector itself Signup and view all the answers

The representation vector depends on the...

Ordering of the elements in the basis Signup and view all the answers

What is the key equation for change of basis?

Signup and view all the answers

What is the transition matrix from U to V?

V inverse times U Signup and view all the answers

What is the transition matrix from V to U?

U inverse times V Signup and view all the answers

Elementary operations do not modify the rowspace of A.

True Signup and view all the answers

Elementary operations do modify the column space of A.

True Signup and view all the answers

What is the nullity of A?

The dimension of the null space of A Signup and view all the answers

What is the rank of A?

The number of pivots in the RREF of A Signup and view all the answers

The number of columns of A equals...

rank(A) + nullity(A) Signup and view all the answers

If the linear system Ax = b is consistent...

b is in the column space of A Signup and view all the answers

If Ax = b is consistent for any b in R^m:

The column span of A is R^m and rank(A) = m Signup and view all the answers

A transformation is linear if and only if:

Signup and view all the answers

When is L used for a transformation versus T?

L is used to emphasize the linearity of a transformation Signup and view all the answers

The image of a linear combination...

Is the linear combination of the images Signup and view all the answers

When checking if a transformation is linear, what should you check first?

If the 0 vector maps - if it does not then we know it is not linear; if it does map then continue by testing steps 1 & 2 Signup and view all the answers

In order to be linear:

The output should be a linear combination of the inputs Signup and view all the answers

A similarity transformation preserves...

Trace and determinant Signup and view all the answers

How do you calculate the trace of a matrix?

Multiply all elements along the diagonal Signup and view all the answers

What does ker(L) equal?

Nullspace (A) Signup and view all the answers

What does range(L) equal?

Columnspace (A) Signup and view all the answers

What is the rotation transformation matrix?

Signup and view all the answers

For clockwise rotations...

Use a negative theta Signup and view all the answers

For counterclockwise rotations...

Use a positive theta Signup and view all the answers

What is the reflection about y=x?

Signup and view all the answers

A matrix is invertible if:

Determinant does not equal 0 Signup and view all the answers

Reflections are always invertible.

True Signup and view all the answers

Rotations are always invertible.

True Signup and view all the answers

Projections are never invertible.

True Signup and view all the answers

Study Notes

Vector Space Basics

A set V consists of a collection of objects, known as vectors.
Operations on set V include "o-plus" (addition) and "o-dot" (multiplication).
Closure conditions ensure that operations produce results within set V and allow for linear combinations.

Axioms of Vector Spaces

A vector space is nonempty and closed under "o-plus" and "o-dot," satisfying axioms A1 to A8 for all x, y, z in V and real numbers alpha, beta.
Axiom 1: "o-plus" is commutative.
Axiom 2: "o-plus" is associative.
Axiom 3: There is a unique zero vector neutral for "o-plus."
Axiom 4: Each vector x has a unique additive inverse -x.
Axiom 5: "o-dot" is associative.
Axiom 6: Distributivity of "o-dot" with respect to the sum of scalars holds.
Axiom 7: Distributivity of multiplication with respect to "o-plus."
Axiom 8: One is the neutral element for "o-dot," akin to the multiplicative identity.

Subspaces and Span

A subset S of V is a subspace if it is closed under "o-plus" and "o-dot," also containing the zero vector.
The span of a set of vectors is the smallest subspace containing all linear combinations of those vectors.
If the span of vectors equals V, they form a spanning set for V.

Linear Independence and Dependence

Vectors are linearly independent if the equation for their linear combination results only in the trivial solution.
A single non-zero vector is always linearly independent, while a set containing the zero vector is linearly dependent.
For two vectors, if one is a scalar multiple of the other, they are linearly dependent.

Basis and Dimension

A basis is a minimal spanning set of linearly independent vectors, uniquely defining the vector space.
The dimension of a vector space is the number of vectors in any basis for that space.
For finite-dimensional spaces, all bases have the same number of vectors.

Transition and Transformation

Transition matrices relate different bases, calculated by multiplying inverse matrices.
Linear transformations preserve the operations of addition and scalar multiplication.
To verify linearity, ensure the zero vector maps correctly and check if linear combinations maintain linearity.

Matrix Properties

A matrix is invertible if its determinant is not zero.
Certain transformations, like rotations and reflections, are always invertible, while projections are not.
The kernel of a transformation corresponds to the nullspace of a matrix, and the range corresponds to the column space.

Special Cases

In R3, the standard basis consists of unit vectors along coordinate axes.
The trace of a matrix is obtained by summing diagonal elements.
For any linear system Ax = b to be consistent, b must reside in the column space of A.

Summary of Key Definitions

Nullspace: Set of solutions for the homogeneous equation Ax = 0.
Rank: Number of leading ones in the row echelon form of a matrix.
The equation relating columns of A includes the rank and nullity as variables balancing the total columns.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge with these flashcards for ASU MAT 343 Exam 2. This quiz covers key concepts such as set theory, operations on sets, and their definitions. Get ready to reinforce your understanding of mathematical operations and their applications in this course.