Understanding Vector Spaces

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Listen to an AI-generated conversation about this lesson
Download our mobile app to listen on the go
Get App

Questions and Answers

Which of the following is an example of a scalar quantity?

  • Temperature (correct)
  • Force
  • Velocity
  • Acceleration

What characteristic distinguishes a vector from a scalar quantity?

  • Vectors have only magnitude.
  • Vectors require both magnitude and direction. (correct)
  • Scalars are always positive.
  • Scalars require a direction.

In the context of vector spaces, what is a 'scalar' primarily used for?

  • Creating a tuple of vectors.
  • Defining the direction of a vector.
  • Scaling the magnitude of a vector during scalar multiplication. (correct)
  • Expressing the magnitude of a vector.

What does the 'length of the arrow' represent in a geometric vector representation?

<p>Magnitude (C)</p>
Signup and view all the answers

What is the correct way to describe the tail of a geometric vector?

<p>Initial Point (B)</p>
Signup and view all the answers

Which of the following statements is true about an ordered n-tuple?

<p>It is a sequence of n real numbers. (B)</p>
Signup and view all the answers

What does $R^n$ represent?

<p>The set of all ordered n-tuples. (B)</p>
Signup and view all the answers

Given two vectors (u = (1, 2)) and (v = (3, 4)), what is the result of (u + v)?

<p>((4, 6)) (C)</p>
Signup and view all the answers

What is the result of scalar multiplication (k * u), where (k = 2) and (u = (3, 1))?

<p>((6, 2)) (A)</p>
Signup and view all the answers

If (u = (u_1, u_2, ..., u_n)) and (v = (v_1, v_2, ..., v_n)) are vectors in (R^n), what is the correct formula for (u + v)?

<p>((u_1 + v_1, u_2 + v_2, ..., u_n + v_n)) (A)</p>
Signup and view all the answers

What condition must a set V satisfy to be considered a real vector space?

<p>It must satisfy ten specific axioms related to vector addition and scalar multiplication. (C)</p>
Signup and view all the answers

Which of the following is NOT one of the axioms that a vector space must satisfy?

<p>Existence of a multiplicative inverse. (B)</p>
Signup and view all the answers

What does the closure under addition axiom state for a vector space V?

<p>For any u, v in V, u + v must also be in V. (C)</p>
Signup and view all the answers

Which property is described by the equation (u + v = v + u) in the context of vector spaces?

<p>Commutative Property (A)</p>
Signup and view all the answers

The equation (u + (v + w) = (u + v) + w) demonstrates which property of vector addition?

<p>Associative (A)</p>
Signup and view all the answers

What is the significance of the 'zero vector' in a vector space?

<p>Adding it to any vector results in the same vector. (A)</p>
Signup and view all the answers

For a vector (u) in a vector space (V), what is its additive inverse?

<p>(-u) (A)</p>
Signup and view all the answers

What does the axiom k(u + v) = ku + kv represent in vector space theory?

<p>Distributive property of scalar multiplication over vector addition (D)</p>
Signup and view all the answers

What is specified by the axiom (k(mu) = (km)u) for a vector space?

<p>Associative property (B)</p>
Signup and view all the answers

In vector spaces, what is the result of multiplying any vector by the scalar '1'?

<p>The original vector (B)</p>
Signup and view all the answers

What defines the 'zero vector space'?

<p>A space that only contains the zero vector. (B)</p>
Signup and view all the answers

Why is (R^n) considered a vector space with standard operations?

<p>Because it is closed under vector addition and scalar multiplication and satisfies all vector space axioms. (A)</p>
Signup and view all the answers

Why are matrices of size 2 2 with real entries considered a vector space?

<p>They are closed under matrix addition and scalar multiplication and satisfy vector space axioms. (A)</p>
Signup and view all the answers

What is a key property that must be satisfied to consider a set (V) a vector space?

<p>The set must be closed under scalar multiplication. (D)</p>
Signup and view all the answers

Which set of polynomials forms a vector space?

<p>Polynomials of degree at most n. (A)</p>
Signup and view all the answers

Consider (V = R^2) with altered scalar multiplication defined as (ku = (ku_1, u_2)). Does (V) still form a vector space?

<p>No, because it fails to satisfy all scalar multiplication axioms. (D)</p>
Signup and view all the answers

Under what condition is (V = R^2) NOT a vector space if scalar multiplication is defined as (k(u, u) = (0, ku))?

<p>Because it does not satisfy the scalar identity axiom. (C)</p>
Signup and view all the answers

If a subset W of a vector space V is itself a vector space under the same operations defined on V, then W is called a:

<p>Subspace (D)</p>
Signup and view all the answers

Which condition is sufficient to prove that a subset (W) of a vector space (V) is a subspace?

<p>If (W) is closed under vector addition and scalar multiplication. (C)</p>
Signup and view all the answers

What is a 'zero subspace'?

<p>A subspace containing only the zero vector. (A)</p>
Signup and view all the answers

Given (W = {(v, 0, v) : v, v R}), is (W) a subspace of (R)?

<p>Yes, because it is closed under addition and scalar multiplication. (C)</p>
Signup and view all the answers

Consider the set W of all points (x, y) in (R^2) where (x 0) and (y 0). Is W a subspace of (R^2)?

<p>No, because it is not closed under scalar multiplication. (B)</p>
Signup and view all the answers

If (W = {(v, v, v) : v + v + v = 0}), is W a subspace of (R)?

<p>Yes, because W satisfies the conditions for being a subspace. (D)</p>
Signup and view all the answers

Let (W = {(v, v, v) : 5v - 3v - 7v = 0}). Is W a subspace of (R)?

<p>Yes, W is a subspace of (R). (D)</p>
Signup and view all the answers

Which of the following sets does NOT form a subspace of (M_{nn}) (the vector space of all n x n matrices)?

<p>Invertible matrices. (D)</p>
Signup and view all the answers

When is the set of all (n x n) matrices with determinant zero NOT a subspace?

<p>Because the set is not closed under addition. (A)</p>
Signup and view all the answers

What is a 'linear combination' of vectors?

<p>The sum of scalar multiples of vectors. (C)</p>
Signup and view all the answers

If (w = kv + kv + ... + k_rv_r), what are the scalars (k, k, ..., k_r) called?

<p>Coefficients (D)</p>
Signup and view all the answers

What does it mean for a set of vectors (S = {w, w, ..., w_r}) to 'span' a vector space (V)?

<p>Every vector in V can be written as a linear combination of vectors in S. (C)</p>
Signup and view all the answers

What is the 'span' of a set (S) of vectors?

<p>The subspace consisting of all possible linear combinations of vectors in (S). (B)</p>
Signup and view all the answers

Standard unit vectors in (R^n) can be best described by?

<p>Span any vector in (R^n) as Linear Combination (C)</p>
Signup and view all the answers

For which condition are the vectors (v, v , ...v_r) in a vector space (V) considered linearly independent?

<p>If the equation (kv + kv + ... + k_rv_r = 0) only has the trivial solution (all (k_i = 0)). (B)</p>
Signup and view all the answers

What characterizes a set of linearly dependent vectors?

<p>At least one of the vectors can be expressed as a linear combination of the others. (A)</p>
Signup and view all the answers

If the only solution to the equation (c_1v_1 + c_2v_2 + c_3v_3 = 0) is (c_1 = c_2 = c_3 = 0), what can be said about the vectors (v_1, v_2), and (v_3)?

<p>They are linearly independent. (B)</p>
Signup and view all the answers

Flashcards

What are scalars?

Quantities described by numerical value alone.

What are vectors?

Quantities needing number and direction for complete description.

What are Geometric Vectors?

Representationof vectors in two or three dimensions using arrows.

What is the initial point of a vector?

The starting location of a geometric vector.

Signup and view all the flashcards

What is the terminal point of a vector?

The end location of a geometric vector (tip of the arrow).

Signup and view all the flashcards

What is an Ordered n-tuple?

Sequence of n real numbers.

Signup and view all the flashcards

What is n-space?

The set of all ordered n-tuples.

Signup and view all the flashcards

What is a Real Vector Space?

A set with ten properties of vector addition & scalar multiplication.

Signup and view all the flashcards

What is Closure Under Addition?

If u, v are in V, then u + v must also be in V.

Signup and view all the flashcards

What is Commutative Property (Addition)?

v + u = u + v for all vectors.

Signup and view all the flashcards

What is Associative Property (Addition)?

(u + v) + w = u + (v + w) for all vectors.

Signup and view all the flashcards

What is the Additive Identity?

There's a zero vector where u + 0 = u.

Signup and view all the flashcards

What is the Additive Inverse?

There exists -u in V, such that u + (-u) = 0.

Signup and view all the flashcards

What is Closure under Scalar Multiplication?

if k is a scalar, ku is in V.

Signup and view all the flashcards

What is the Distributive Property?

k(u + v) = ku + kv for a scalar k.

Signup and view all the flashcards

What is the Distributive Property?

(k + m)u = ku + mu for scalars k, m.

Signup and view all the flashcards

What is the Associative Property(Scalar Mult)?

k(mu) = (km)u for scalars k, m.

Signup and view all the flashcards

What is the Scalar Identity?

1u = u for all vectors u.

Signup and view all the flashcards

What is a Subspace?

A subset of a vector space that is also a vector space.

Signup and view all the flashcards

What is Closure under addition?

If u and v are in W, then u + v is also in W.

Signup and view all the flashcards

What is Closure under scalar multiplication?

If k is a scalar and u is in W, then ku is in W.

Signup and view all the flashcards

What is the Zero Subspace?

Subset containing zero vector only; a subspace of V

Signup and view all the flashcards

What is Closure Under Additon?

Adding vectors yields another vector in set.

Signup and view all the flashcards

What is Closure Under Scalar Multiplication?

Multiplying by scalar yields vector in set.

Signup and view all the flashcards

What is a Linear Combination?

w can be expressed using scalars k and vectors v.

Signup and view all the flashcards

What are Coefficients?

Scalars multiplied to vectors in a linear combination.

Signup and view all the flashcards

What is Span?

Set that contains all possible linear combinations of vectors.

Signup and view all the flashcards

What is Span(S)?

A set is a subspace of V.

Signup and view all the flashcards

What are Standard Unit Vectors?

e₁ = (1,0, ..., 0), e₂ = (0,1, ..., 0), ..., en = (0,0, ..., 1)

Signup and view all the flashcards

What is a Linearly Independent Set?

If no vector in S is a combination of others.

Signup and view all the flashcards

What is a Linearly Dependent Set?

A set that is not linearly independent.

Signup and view all the flashcards

Linearly Independent Set?

All coefficients must be zero.

Signup and view all the flashcards

Study Notes

Vector Spaces

  • Engineers and physicists use two types of physical quantities: scalars and vectors.
  • Scalars have just magnitude
  • Vectors have magnitude and direction
  • Vectors and scalars notions can be used in genetics, computer science, economics, telecommunications, and environmental science.

Geometric Vectors

  • Vectors are represented in two dimensions, known as 2-space, or in three dimensions, known as 3-space, using arrows.
  • The direction of an arrowhead shows the vector's direction.
  • The length of the arrow shows its magnitude.
  • The tail of the arrow is the vector's initial point.
  • The tip of the arrow is the terminal point.
  • An ordered n-tuple is a sequence of n real numbers represented as (v1, v2, ..., vn), where n is a positive integer
  • The n-space set is the set of all ordered n-tuples, and is denoted by Rn.

Vector Representation

  • Vectors can be represented as a 1 × n row matrix or an n × 1 column matrix.
  • Matrix addition and scalar multiplication produce the same results as corresponding vector operations.

Real Vector Spaces

  • A set that fulfills ten properties of vector addition and scalar multiplication in Rn, or axioms, is a vector space.
  • Its set objects are vectors
  • V is a vector space if it is a set in which two operations (vector addition and scalar multiplication) are defined.
  • Given that the listed axioms are fulfilled for every u, v, and w in V and every scalar (real number) k and m

Addition Axioms:

  • Closure under addition: if u and v are in V, then u + v is in V.
  • Commutative property: u + v = v + u
  • Associative property: u + (v + w) = (u + v) + w
  • Additive identity: V has a zero vector 0 such that for any u in V, 0 + u = u + 0 = u.
  • Additive inverse: for each u in V, there exists -u in V such that u + (-u) = (-u) + u = 0.

Scalar Multiplication Axioms:

  • Closure under scalar multiplication: if k is a scalar and u is in V, then ku is in V.
  • Distributive property: k(u + v) = ku + kv
  • Distributive property: (k + m)u = ku + mu
  • Associative property: k(mu) = (km)(u)
  • Scalar identity: 1u = u

The Zero Vector Space Example

  • If V consists of just a zero object, defining 0 + 0 = 0 and k0 = 0 for all scalars k fulfills all vector space axioms.

Rn with Standard Operations

  • Defining vector space operations on V to be the standard addition and scalar multiplication of n-tuples means that:
    • u + v = (u1, ..., un) + (v1, ..., vn) = (u₁ + v₁, ..., un + vn)
    • ku = (ku₁, ..., kun)
  • V = Rn is closed under scalar multiplication and addition, produces n-tuples as their end result and meets all Axioms.

Matrices 2 × 2

  • Considering V as the set of 2 × 2 matrices with real entries
  • u + v and ku operations work as expected
  • All 2 × 2 matrices are a vector space

Polynomials Vector Space Example

  • For n ≥ 0, the set Pn of polynomials with a degree of at most n includes all polynomials in the form P(t) = a₀ + a₁t + ... + antn, with coefficients a₀, ..., an and variable t as real numbers.
  • The degree of P is the highest power of t in the polynomial with a nonzero coefficient,
  • If all coefficients are zero, P is the zero polynomial. The zero polynomial definition satisfies axioms 1 through 10.
  • Pn is a vector space.

Vector Space Example

  • V = R2, where addition is defined as u + v = (u₁ + v₁, u₂ + v₂), and scalar multiplication is defined as ku = (ku₁, u₂)
  • The addition operation is the standard one from R2.
  • The scalar multiplication is not and Axiom 5 fails to hold
  • Thus, V is not a vector space

Subspaces

  • If subset W of vector space V is a subspace, then W is a vector space with the addition and scalar multiplication process defined on V.
  • W is a subspace of V if the vectors are in a vector space V and meet the following conditions:
    • If u and v are vectors in W, then u + v is in W.
    • If k is a scalar and u is a vector in W, then ku is in W.

The Zero Subspace

  • If V is any vector space and W = {0} is the subset of V with just the zero vector, W is closed under addition and scalar multiplication.
  • Thus, W is the zero subspace of V.

Vector Space Subspaces

  • Set W = {(v1, 0, v3): v1, v3 ∈ R} is a subspace of R3 with standard operations: if u and v in W then u+v is in W
  • Set W = {(v1, v2, 0): v1, v2 ∈ R} is a subspace of R3 with standard operations: if u and v in W then u+v is in W

Non Subspaces

  • For the set W of all points (x, y) in R2 with x ≥ 0 and y ≥ 0, W is not a subspace of R2 because it is not closed under scalar multiplication.

Determining Subspaces In Vector Spaces

  • W = {(v1, v2, v3): v1 + v2 + v3 = 0 ∈ R} subspace of R3: u and v in W, then u+v is in W, also ku is in W, W is a subspace of vector space V
  • Set W = {(v1, v2, v3): 5v1 - 3v2 - 7v3 = 0 ∈ R} , then W is a subspace of vector space V
  • If W = {x ∈ R2: x = [2t,t], with t any real number}, then W is a subspace of vector space V
  • The invertible n × n matrices set W is not a subspace of Mnn, failing on two counts: not closed under addition and scalar multiplication.
  • Also W is not a subspace of R³
  • However Is the set W is a subspace of R3
  • W is a subspace of Mnn if it is all symmetric, upper triangular, lower triangular and diagonal matrices
  • For W = {M2x2: det(A) = 0}, W is not a subspace of M2×2, however when W = {M3×3: tr(A) = 0}, W is a subspace of M3×3
  • the set W = {v: v3 − v3 = 0} is not subspace of R², as it doesn't meet the criteria for being a subspace.

Spanning Sets and Linear Independence

  • A vector w in a vector space V is a linear combination of vectors v1, v2, ..., vr in V if it can be expressed as w = k1v1 + k2v2 + ... + krvr, where k1, k2, ..., kr are scalars called coefficients.
  • Span(S) is called the subspace of V generated by S, and can be expressed as W = span{w1,w2,..., wr} or W = span(S)

Theorem Span(S) Is a Subspace of V

  • Span(S) is a subspace of V that contains S and every subspace of V containing S, Span(S) must contain

Standard Unit Vectors

  • Span Rn, the Standard Unit Vectors in Rn are e₁ = (1,0, ..., 0), e₂ = (0,1, ..., 0), ..., en = (0,0, ..., 1)
  • Every vector v expressed as (V1, V2, ..., vn) can be expressed as v = v₁e₁ + v₂e2+ ... + vnen

Properties of Spanning Sets

  • Sets can be used for spanning sets of R2 or R3

Linear Independence and Dependence

  • The set S, which contains two or more vectors in vector space V, is linearly independent if no vector in S can be expressed as the others linear combination.
  • The opposite is a linearly dependent set
  • The set S in a vector space V is linearly independent if the only coefficients is the vector equation, shown as k₁v1 + k2v2 + ··· + krvr = 0, are k₁ = 0, k2 = 0, ..., kr = 0.
  • If vectors are independent, the only solution is 0,0. If they are dependent then vectors are not scalar multiples of each others
  • Vectors are linearly dependent in Rn, because an homogeneous system has an underdetermined variable leading to solutions. Also v1, v2 and v3 are linearly dependent if the determinant is 0.

Studying That Suits You

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

Quiz Team

Related Documents

Vector Spaces PDF

More Like This

Exploring Vector Spaces in Linear Algebra
10 questions
Linear Algebra: Vector Spaces
10 questions
Use Quizgecko on...
Browser
Browser