Podcast
Questions and Answers
Which of the following is an example of a scalar quantity?
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?
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?
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?
What does the 'length of the arrow' represent in a geometric vector representation?
What is the correct way to describe the tail of a geometric vector?
What is the correct way to describe the tail of a geometric vector?
Which of the following statements is true about an ordered n-tuple?
Which of the following statements is true about an ordered n-tuple?
What does $R^n$ represent?
What does $R^n$ represent?
Given two vectors (u = (1, 2)) and (v = (3, 4)), what is the result of (u + v)?
Given two vectors (u = (1, 2)) and (v = (3, 4)), what is the result of (u + v)?
What is the result of scalar multiplication (k * u), where (k = 2) and (u = (3, 1))?
What is the result of scalar multiplication (k * u), where (k = 2) and (u = (3, 1))?
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)?
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)?
What condition must a set V satisfy to be considered a real vector space?
What condition must a set V satisfy to be considered a real vector space?
Which of the following is NOT one of the axioms that a vector space must satisfy?
Which of the following is NOT one of the axioms that a vector space must satisfy?
What does the closure under addition axiom state for a vector space V?
What does the closure under addition axiom state for a vector space V?
Which property is described by the equation (u + v = v + u) in the context of vector spaces?
Which property is described by the equation (u + v = v + u) in the context of vector spaces?
The equation (u + (v + w) = (u + v) + w) demonstrates which property of vector addition?
The equation (u + (v + w) = (u + v) + w) demonstrates which property of vector addition?
What is the significance of the 'zero vector' in a vector space?
What is the significance of the 'zero vector' in a vector space?
For a vector (u) in a vector space (V), what is its additive inverse?
For a vector (u) in a vector space (V), what is its additive inverse?
What does the axiom k(u + v) = ku + kv
represent in vector space theory?
What does the axiom k(u + v) = ku + kv
represent in vector space theory?
What is specified by the axiom (k(mu) = (km)u) for a vector space?
What is specified by the axiom (k(mu) = (km)u) for a vector space?
In vector spaces, what is the result of multiplying any vector by the scalar '1'?
In vector spaces, what is the result of multiplying any vector by the scalar '1'?
What defines the 'zero vector space'?
What defines the 'zero vector space'?
Why is (R^n) considered a vector space with standard operations?
Why is (R^n) considered a vector space with standard operations?
Why are matrices of size 2 2 with real entries considered a vector space?
Why are matrices of size 2 2 with real entries considered a vector space?
What is a key property that must be satisfied to consider a set (V) a vector space?
What is a key property that must be satisfied to consider a set (V) a vector space?
Which set of polynomials forms a vector space?
Which set of polynomials forms a vector space?
Consider (V = R^2) with altered scalar multiplication defined as (ku = (ku_1, u_2)). Does (V) still form a vector space?
Consider (V = R^2) with altered scalar multiplication defined as (ku = (ku_1, u_2)). Does (V) still form a vector space?
Under what condition is (V = R^2) NOT a vector space if scalar multiplication is defined as (k(u, u) = (0, ku))?
Under what condition is (V = R^2) NOT a vector space if scalar multiplication is defined as (k(u, u) = (0, ku))?
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:
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:
Which condition is sufficient to prove that a subset (W) of a vector space (V) is a subspace?
Which condition is sufficient to prove that a subset (W) of a vector space (V) is a subspace?
What is a 'zero subspace'?
What is a 'zero subspace'?
Given (W = {(v, 0, v) : v, v R}), is (W) a subspace of (R)?
Given (W = {(v, 0, v) : v, v R}), is (W) a subspace of (R)?
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)?
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)?
If (W = {(v, v, v) : v + v + v = 0}), is W a subspace of (R)?
If (W = {(v, v, v) : v + v + v = 0}), is W a subspace of (R)?
Let (W = {(v, v, v) : 5v - 3v - 7v = 0}). Is W a subspace of (R)?
Let (W = {(v, v, v) : 5v - 3v - 7v = 0}). Is W a subspace of (R)?
Which of the following sets does NOT form a subspace of (M_{nn}) (the vector space of all n x n matrices)?
Which of the following sets does NOT form a subspace of (M_{nn}) (the vector space of all n x n matrices)?
When is the set of all (n x n) matrices with determinant zero NOT a subspace?
When is the set of all (n x n) matrices with determinant zero NOT a subspace?
What is a 'linear combination' of vectors?
What is a 'linear combination' of vectors?
If (w = kv + kv + ... + k_rv_r), what are the scalars (k, k, ..., k_r) called?
If (w = kv + kv + ... + k_rv_r), what are the scalars (k, k, ..., k_r) called?
What does it mean for a set of vectors (S = {w, w, ..., w_r}) to 'span' a vector space (V)?
What does it mean for a set of vectors (S = {w, w, ..., w_r}) to 'span' a vector space (V)?
What is the 'span' of a set (S) of vectors?
What is the 'span' of a set (S) of vectors?
Standard unit vectors in (R^n) can be best described by?
Standard unit vectors in (R^n) can be best described by?
For which condition are the vectors (v, v , ...v_r) in a vector space (V) considered linearly independent?
For which condition are the vectors (v, v , ...v_r) in a vector space (V) considered linearly independent?
What characterizes a set of linearly dependent vectors?
What characterizes a set of linearly dependent vectors?
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)?
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)?
Flashcards
What are scalars?
What are scalars?
Quantities described by numerical value alone.
What are vectors?
What are vectors?
Quantities needing number and direction for complete description.
What are Geometric Vectors?
What are Geometric Vectors?
Representationof vectors in two or three dimensions using arrows.
What is the initial point of a vector?
What is the initial point of a vector?
The starting location of a geometric vector.
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What is the terminal point of a vector?
What is the terminal point of a vector?
The end location of a geometric vector (tip of the arrow).
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What is an Ordered n-tuple?
What is an Ordered n-tuple?
Sequence of n real numbers.
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What is n-space?
What is n-space?
The set of all ordered n-tuples.
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What is a Real Vector Space?
What is a Real Vector Space?
A set with ten properties of vector addition & scalar multiplication.
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What is Closure Under Addition?
What is Closure Under Addition?
If u, v are in V, then u + v must also be in V.
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What is Commutative Property (Addition)?
What is Commutative Property (Addition)?
v + u = u + v for all vectors.
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What is Associative Property (Addition)?
What is Associative Property (Addition)?
(u + v) + w = u + (v + w) for all vectors.
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What is the Additive Identity?
What is the Additive Identity?
There's a zero vector where u + 0 = u.
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What is the Additive Inverse?
What is the Additive Inverse?
There exists -u in V, such that u + (-u) = 0.
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What is Closure under Scalar Multiplication?
What is Closure under Scalar Multiplication?
if k is a scalar, ku is in V.
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What is the Distributive Property?
What is the Distributive Property?
k(u + v) = ku + kv for a scalar k.
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What is the Distributive Property?
What is the Distributive Property?
(k + m)u = ku + mu for scalars k, m.
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What is the Associative Property(Scalar Mult)?
What is the Associative Property(Scalar Mult)?
k(mu) = (km)u for scalars k, m.
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What is the Scalar Identity?
What is the Scalar Identity?
1u = u for all vectors u.
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What is a Subspace?
What is a Subspace?
A subset of a vector space that is also a vector space.
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What is Closure under addition?
What is Closure under addition?
If u and v are in W, then u + v is also in W.
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What is Closure under scalar multiplication?
What is Closure under scalar multiplication?
If k is a scalar and u is in W, then ku is in W.
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What is the Zero Subspace?
What is the Zero Subspace?
Subset containing zero vector only; a subspace of V
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What is Closure Under Additon?
What is Closure Under Additon?
Adding vectors yields another vector in set.
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What is Closure Under Scalar Multiplication?
What is Closure Under Scalar Multiplication?
Multiplying by scalar yields vector in set.
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What is a Linear Combination?
What is a Linear Combination?
w can be expressed using scalars k and vectors v.
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What are Coefficients?
What are Coefficients?
Scalars multiplied to vectors in a linear combination.
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What is Span?
What is Span?
Set that contains all possible linear combinations of vectors.
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What is Span(S)?
What is Span(S)?
A set is a subspace of V.
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What are Standard Unit Vectors?
What are Standard Unit Vectors?
e₁ = (1,0, ..., 0), e₂ = (0,1, ..., 0), ..., en = (0,0, ..., 1)
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What is a Linearly Independent Set?
What is a Linearly Independent Set?
If no vector in S is a combination of others.
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What is a Linearly Dependent Set?
What is a Linearly Dependent Set?
A set that is not linearly independent.
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Linearly Independent Set?
Linearly Independent Set?
All coefficients must be zero.
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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.
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