Linear Algebra: Scalars and Vectors

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Questions and Answers

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?

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?

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?

Adding two vectors results in another vector of the same kind. Signup and view all the answers

What is a characteristic of vectors that distinguishes them from scalars?

Vectors are geometric objects with both magnitude and direction. Signup and view all the answers

Which mathematical property is illustrated by the equation 𝒓 + 𝒔 = 𝒔 + 𝒓?

Commutative property Signup and view all the answers

In the context of vectors, what does the associative property imply?

The grouping of vectors does not change the result of their addition. Signup and view all the answers

Which statement correctly identifies how vectors can be manipulated mathematically?

Vectors can be multiplied by scalars to produce another vector. Signup and view all the answers

What is the result of the dot product of vectors a=(2,3,1) and b=(4,−1,2)?

7 Signup and view all the answers

What geometric interpretation does a dot product greater than zero signify?

Angle between vectors is acute. Signup and view all the answers

Which property of the dot product states that r.s = s.r?

Commutative property Signup and view all the answers

Which situation describes vectors being orthogonal?

Dot product equals zero. Signup and view all the answers

What does the equation cos(θ) = a⋅b / |a||b| help to calculate?

Angle between two vectors. Signup and view all the answers

What dimensional requirement exists for the vectors when calculating the dot product?

Vectors must be of equal dimensions. Signup and view all the answers

When calculating the modulus of a vector, r.r results in which expression?

|r|^2 Signup and view all the answers

Which application does the dot product NOT relate to?

Finding volume of a shape. Signup and view all the answers

What is the result of adding the vectors (2, 2, 0) and (3, 3, 0)?

(5, 5, 0) Signup and view all the answers

How can a vector be scaled by a scalar $oldsymbol{oldsymbol{ ho}}$?

Multiply each element by $ ho$ Signup and view all the answers

Which of the following defines a vector space?

A set of vectors formed by scaling and adding any two vectors Signup and view all the answers

If a polynomial is represented as a vector, what happens when two polynomials are added?

A third polynomial is generated Signup and view all the answers

Which statement about vector dimensions is accurate?

A vector can be represented as a single dimension in addition to others Signup and view all the answers

What is the modulus of a vector?

The length or magnitude of the vector Signup and view all the answers

Given the vector $oldsymbol{r} = (3, 5, -3)$, what is $−oldsymbol{r}$?

(−3, -5, 3) Signup and view all the answers

In data science, which example best illustrates the features included in a vector?

Each feature can represent multiple attributes of a dataset Signup and view all the answers

What is the modulus of the vector represented by the components 3 and 4?

$5$ Signup and view all the answers

In a multidimensional vector, how is the modulus calculated?

Using the formula $\sqrt{a_1^2 + a_2^2 + ... + a_n^2}$ Signup and view all the answers

What is a unit vector?

A vector with a modulus equal to 1 Signup and view all the answers

What does the equation $c^2 = a^2 + b^2 - 2ab , cos \theta$ represent?

The cosine rule in a triangle Signup and view all the answers

Why is normalizing a vector useful?

It simplifies the vector to unit length Signup and view all the answers

If the angle $ heta$ between two vectors is 0°, what is the relationship between their dot product and magnitudes?

The dot product equals the product of their magnitudes Signup and view all the answers

What value does the dot product of two orthogonal vectors yield?

0 Signup and view all the answers

Which statement about the dot product is true?

It is the sum of the products of components in the same dimension Signup and view all the answers

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?

It is the angle between the vectors r and s Signup and view all the answers

If vector a has components (0, 5) and vector b has components (3, 4), which of the following is true?

|a + b| ≤ |a| + |b| Signup and view all the answers

What happens to the dot product of two vectors if they are pointing in opposite directions?

The dot product becomes negative Signup and view all the answers

What is the result of normalizing a vector (4, 3)?

(0.8, 0.6) Signup and view all the answers

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?

$\frac{5}{\sqrt{406}}$ Signup and view all the answers

Which inequality represents the relationship between the modulus of a sum of vectors and the sum of their moduli?

|a + b| ≤ |a| + |b| Signup and view all the answers

What is the result when calculating $r \cdot s$ if $ heta = 90°$?

It equals zero Signup and view all the answers

What is a necessary condition for two vectors $ extbf{r}$ and $ extbf{s}$ to satisfy the relationship $\mathbf{r} \mathbf{s} \cos \theta = \mathbf{r} \cdot \mathbf{s}$?

They can be any vectors Signup and view all the answers

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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Description

This quiz covers the foundational concepts of linear algebra, focusing on scalars and vectors. Understand the definitions, operations, and properties that govern these elements in both mathematics and physics. Test your knowledge on vector addition, scaling, and their graphical representations.