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Questions and Answers
What defines a scalar in the context of linear algebra?
What defines a scalar in the context of linear algebra?
- A single value, such as 7 or -4 (correct)
- A quantity represented by an arrow in space
- A single variable with both magnitude and direction
- A mathematical object that can only be added to another scalar
What is a key characteristic of vectors as described in the physics way?
What is a key characteristic of vectors as described in the physics way?
- They consist of only one element or variable.
- They cannot be represented graphically.
- They possess both magnitude and direction. (correct)
- They are quantities that can be only multiplied by scalars.
In linear algebra, which statement about vector addition is true?
In linear algebra, which statement about vector addition is true?
- Vectors can be added only if they are of different sizes.
- Vectors cannot be multiplied together.
- Vector addition is always commutative. (correct)
- Vector addition is reversible but not associative.
Which of the following best describes the mathematical representation of vector addition?
Which of the following best describes the mathematical representation of vector addition?
What is a characteristic of vectors that distinguishes them from scalars?
What is a characteristic of vectors that distinguishes them from scalars?
Which mathematical property is illustrated by the equation 𝒓 + 𝒔 = 𝒔 + 𝒓?
Which mathematical property is illustrated by the equation 𝒓 + 𝒔 = 𝒔 + 𝒓?
In the context of vectors, what does the associative property imply?
In the context of vectors, what does the associative property imply?
Which statement correctly identifies how vectors can be manipulated mathematically?
Which statement correctly identifies how vectors can be manipulated mathematically?
What is the result of the dot product of vectors a=(2,3,1) and b=(4,−1,2)?
What is the result of the dot product of vectors a=(2,3,1) and b=(4,−1,2)?
What geometric interpretation does a dot product greater than zero signify?
What geometric interpretation does a dot product greater than zero signify?
Which property of the dot product states that r.s = s.r?
Which property of the dot product states that r.s = s.r?
Which situation describes vectors being orthogonal?
Which situation describes vectors being orthogonal?
What does the equation cos(θ) = a⋅b / |a||b| help to calculate?
What does the equation cos(θ) = a⋅b / |a||b| help to calculate?
What dimensional requirement exists for the vectors when calculating the dot product?
What dimensional requirement exists for the vectors when calculating the dot product?
When calculating the modulus of a vector, r.r results in which expression?
When calculating the modulus of a vector, r.r results in which expression?
Which application does the dot product NOT relate to?
Which application does the dot product NOT relate to?
What is the result of adding the vectors (2, 2, 0) and (3, 3, 0)?
What is the result of adding the vectors (2, 2, 0) and (3, 3, 0)?
How can a vector be scaled by a scalar $oldsymbol{oldsymbol{
ho}}$?
How can a vector be scaled by a scalar $oldsymbol{oldsymbol{ ho}}$?
Which of the following defines a vector space?
Which of the following defines a vector space?
If a polynomial is represented as a vector, what happens when two polynomials are added?
If a polynomial is represented as a vector, what happens when two polynomials are added?
Which statement about vector dimensions is accurate?
Which statement about vector dimensions is accurate?
What is the modulus of a vector?
What is the modulus of a vector?
Given the vector $oldsymbol{r} = (3, 5, -3)$, what is $−oldsymbol{r}$?
Given the vector $oldsymbol{r} = (3, 5, -3)$, what is $−oldsymbol{r}$?
In data science, which example best illustrates the features included in a vector?
In data science, which example best illustrates the features included in a vector?
What is the modulus of the vector represented by the components 3 and 4?
What is the modulus of the vector represented by the components 3 and 4?
In a multidimensional vector, how is the modulus calculated?
In a multidimensional vector, how is the modulus calculated?
What is a unit vector?
What is a unit vector?
What does the equation $c^2 = a^2 + b^2 - 2ab , cos \theta$ represent?
What does the equation $c^2 = a^2 + b^2 - 2ab , cos \theta$ represent?
Why is normalizing a vector useful?
Why is normalizing a vector useful?
If the angle $ heta$ between two vectors is 0°, what is the relationship between their dot product and magnitudes?
If the angle $ heta$ between two vectors is 0°, what is the relationship between their dot product and magnitudes?
What value does the dot product of two orthogonal vectors yield?
What value does the dot product of two orthogonal vectors yield?
Which statement about the dot product is true?
Which statement about the dot product is true?
For the equation $|\mathbf{r} - \mathbf{s}|^2 = |\mathbf{r}|^2 + |\mathbf{s}|^2 - 2 \mathbf{r} \mathbf{s} , cos \theta$, how is $ heta$ related to the vectors?
For the equation $|\mathbf{r} - \mathbf{s}|^2 = |\mathbf{r}|^2 + |\mathbf{s}|^2 - 2 \mathbf{r} \mathbf{s} , cos \theta$, how is $ heta$ related to the vectors?
If vector a has components (0, 5) and vector b has components (3, 4), which of the following is true?
If vector a has components (0, 5) and vector b has components (3, 4), which of the following is true?
What happens to the dot product of two vectors if they are pointing in opposite directions?
What happens to the dot product of two vectors if they are pointing in opposite directions?
What is the result of normalizing a vector (4, 3)?
What is the result of normalizing a vector (4, 3)?
Given two vectors with magnitudes $|a| = \sqrt{29}$ and $|b| = \sqrt{14}$, if their dot product is 5, what is the cosine of the angle between them?
Given two vectors with magnitudes $|a| = \sqrt{29}$ and $|b| = \sqrt{14}$, if their dot product is 5, what is the cosine of the angle between them?
Which inequality represents the relationship between the modulus of a sum of vectors and the sum of their moduli?
Which inequality represents the relationship between the modulus of a sum of vectors and the sum of their moduli?
What is the result when calculating $r \cdot s$ if $ heta = 90°$?
What is the result when calculating $r \cdot s$ if $ heta = 90°$?
What is a necessary condition for two vectors $ extbf{r}$ and $ extbf{s}$ to satisfy the relationship $\mathbf{r} \mathbf{s} \cos \theta = \mathbf{r} \cdot \mathbf{s}$?
What is a necessary condition for two vectors $ extbf{r}$ and $ extbf{s}$ to satisfy the relationship $\mathbf{r} \mathbf{s} \cos \theta = \mathbf{r} \cdot \mathbf{s}$?
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Study Notes
Linear Algebra
- Linear algebra studies vectors and the rules that we need to manipulate these vectors
- Algebra is a set of symbols and rules that formalise concepts
Scalars
- A scalar is a single value, denoted by x
- Examples of scalars are 7, -4, and 1/3
Vectors
- Vectors are quantities that can be added together and multiplied by scalars to produce another object of the same kind.
- Two vectors can be added together 𝒓 + 𝒔 = 𝒕, where 𝒕 is another vector.
- Vectors that are added must have the same number of elements, also called size.
Vectors: Physics Perspective
- Vectors are arrows pointing in space
- Vectors have magnitude and direction
Vectors: Maths Perspective
- Vectors can be added together graphically
- Adding vectors is commutative: 𝒓 + 𝒔 = 𝒔 + 𝒓
- Adding vectors is associative: 𝒓+𝒔 +𝒕 = 𝒓+ 𝒔+𝒕
- The additive inverse of a vector is 𝒓 + −𝒓 = 𝟎
- Vectors can be scaled by a number, 𝜆∈ℝ, where 𝜆 is the scalar: 𝜆𝒙, 𝟑𝒓, −𝒓
- To scale a vector, multiply each element by the scalar.
Vectors: Data Science Perspective
- Vectors are ordered lists of numbers
- Vectors are tuples of 𝑛 real numbers arranged in a row or column, 𝒙 or 𝑥Ԧ
- A row vector is 5 −2 8
- A column vector is 9 1 7 -3
- Each element of a vector represents a feature or dimension.
Vector Space
- The set of vectors that we get by scaling and/or adding any two vectors is called the vector space.
Vector Modulus
- The modulus of a vector is its magnitude.
- The modulus of vector 3 4 is √(3^2)+(4^2) = 5
- The modulus of a vector 𝒓 = 𝑎𝑖Ƹ + 𝑏𝒋Ƹ is √(𝑎^2)+(𝑏^2)
Vector Modulus: Multidimensional Vectors
- The modulus of a vector that has more than 2 dimensions, 𝒓 = 𝑎1 ⋮ 𝑎𝑛, is √(𝑎1^2) + ⋯ + (𝑎𝑛^2)
The Unit Vector
- A unit vector has a modulus of 1
- Normalize a vector by dividing all the vector elements by the vector modulus.
The Dot Product (Inner Product or Projection Product)
- The dot product of vectors 𝒓 and 𝒔 is denoted by 𝒓.𝒔 and is given by 𝒓𝑁 𝒔 = σ𝑁 𝑖=1 𝑟𝑖 𝑠𝑖
- The dot product is the sum of the products of the values in the same dimension.
- If 𝒓.𝒔 > 0 then the angle between the vectors is acute (< 90°), the vectors point roughly in the same direction.
- If 𝒓.𝒔 = 0 then the vectors are orthogonal (perpendicular, 90° angle) and are independent in direction.
- If 𝒓.𝒔 < 0 then the angle between the vectors is obtuse (> 90°), the vectors point roughly in opposite directions.
Dot Product: Properties
- The dot product can only be applied on vectors of equal dimensions
- The dot product always returns a scalar.
- The dot product is commutative: 𝒓.𝒔 = 𝒔.𝒓
- The dot product is distributive over addition: 𝒓.𝒔 + 𝒕 = 𝒓.𝒔 + 𝒓.𝒕
- The dot product is associative over scalar multiplication: 𝒓.𝑎𝒕 = 𝑎(𝒓.𝒕)
Dot Product: Calculating Vector Modulus
- 𝒓.𝒓 = 𝑟1 𝑟1 + 𝑟2 𝑟2 = 𝑟1 2 + 𝑟2 2 = 2 𝑟1 2 + 𝑟2 2 = |𝒓|2
- Therefore, 𝒓.𝒓 = | 𝒓 |
Dot Product: Cosine Rule
- |𝒓 − 𝒔|2 = |𝒓|2 + |𝒔|2 − 2 𝒓 𝒔 cos 𝜃
- 𝒓.𝒔 cos 𝜃 = 𝒓.𝒔
Dot Product: Further Insights
- If 𝜃 = 0° then 𝒓.𝒔 = 𝒓 |𝒔|
- If 𝜃 = 90° then 𝒓.𝒔 = 0, the vectors are orthogonal
- If vectors point in opposite directions then 𝒓.𝒔 = − 𝒓 |𝒔|
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