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Mathematics for Data Science

bookIntroductions to Vectors

Note
Definition

A vector is a mathematical object that represents both direction and magnitude in space. In data science, vectors are used to describe data points, features, and model parameters such as weights.

What Is a Vector?

A vector is an ordered pair of numbers with both magnitude and direction.

v⃗=(x,y)\vec{v} = (x,y)

Vectors are often drawn as arrows from the origin to a point in space. Two vectors are considered equal if they have the same direction and length, even if they start at different locations.

The Zero Vector

The zero vector has no length and no direction. It is written as:

0βƒ—=(0,0)\vec{0} = (0, 0)

Vector Addition and Subtraction

Addition

To add two vectors, add their corresponding components:

aβƒ—+bβƒ—=(a1+b1,β€…β€Ša2+b2)\vec{a} + \vec{b} = (a_1 + b_1, \; a_2 + b_2)

You can visualize this with:

  • Head-to-tail method: move the tail of one vector to the head of the other;
  • Parallelogram method: both vectors start from the same point and form a parallelogram.

Subtraction

To subtract one vector from another:

aβƒ—βˆ’bβƒ—=(a1βˆ’b1,β€…β€Ša2βˆ’b2)\vec{a} - \vec{b} = (a_1 - b_1, \; a_2 - b_2)

This gives a new vector pointing from the head of the second to the head of the first.

Scalar Multiplication

Multiplying a vector by a number (a scalar) stretches or flips the vector:

kβ‹…aβƒ—=(kβ‹…a1,β€…β€Škβ‹…a2)k \cdot \vec{a} = (k \cdot a_1, \; k \cdot a_2)
  • If k>1k > 1, the vector is stretched in the same direction;
  • If 0<k<10 < k < 1, the vector is shrunk;
  • If k<0k < 0, it flips direction;
  • If k=0k = 0, it becomes the zero vector.

Vector Magnitude (Length)

The magnitude or length of a vector is calculated with the Pythagorean theorem:

∣aβƒ—βˆ£=a12+a22|\vec{a}| = \sqrt{a_1^2 + a_2^2}

This gives the straight-line distance from the origin to the tip of the vector.

The Dot Product

The dot product combines two vectors into a single number that reflects how aligned they are:

a⃗⋅b⃗=a1b1+a2b2\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2
  • If the result is positive: the vectors point in a similar direction;
  • If the result is zero: the vectors are perpendicular;
  • If the result is negative: they point in opposite directions.

Example

If a⃗=(1,2)  and  b⃗=(3,4) \vec{a} = (1, 2)\ \ \text{and}\ \ \vec{b} = (3, 4), then:

a⃗⋅b⃗=1⋅3+2⋅4=11\vec{a} \cdot \vec{b} = 1 \cdot 3 + 2 \cdot 4 = 11
question mark

If a⃗=(1,0), b⃗=(0,1)\vec{a} = (1, 0),\ \vec{b} = (0, 1). Then their dot product is:

Select the correct answer

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How can we improve it?

Thanks for your feedback!

SectionΒ 4. ChapterΒ 1

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bookIntroductions to Vectors

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Note
Definition

A vector is a mathematical object that represents both direction and magnitude in space. In data science, vectors are used to describe data points, features, and model parameters such as weights.

What Is a Vector?

A vector is an ordered pair of numbers with both magnitude and direction.

v⃗=(x,y)\vec{v} = (x,y)

Vectors are often drawn as arrows from the origin to a point in space. Two vectors are considered equal if they have the same direction and length, even if they start at different locations.

The Zero Vector

The zero vector has no length and no direction. It is written as:

0βƒ—=(0,0)\vec{0} = (0, 0)

Vector Addition and Subtraction

Addition

To add two vectors, add their corresponding components:

aβƒ—+bβƒ—=(a1+b1,β€…β€Ša2+b2)\vec{a} + \vec{b} = (a_1 + b_1, \; a_2 + b_2)

You can visualize this with:

  • Head-to-tail method: move the tail of one vector to the head of the other;
  • Parallelogram method: both vectors start from the same point and form a parallelogram.

Subtraction

To subtract one vector from another:

aβƒ—βˆ’bβƒ—=(a1βˆ’b1,β€…β€Ša2βˆ’b2)\vec{a} - \vec{b} = (a_1 - b_1, \; a_2 - b_2)

This gives a new vector pointing from the head of the second to the head of the first.

Scalar Multiplication

Multiplying a vector by a number (a scalar) stretches or flips the vector:

kβ‹…aβƒ—=(kβ‹…a1,β€…β€Škβ‹…a2)k \cdot \vec{a} = (k \cdot a_1, \; k \cdot a_2)
  • If k>1k > 1, the vector is stretched in the same direction;
  • If 0<k<10 < k < 1, the vector is shrunk;
  • If k<0k < 0, it flips direction;
  • If k=0k = 0, it becomes the zero vector.

Vector Magnitude (Length)

The magnitude or length of a vector is calculated with the Pythagorean theorem:

∣aβƒ—βˆ£=a12+a22|\vec{a}| = \sqrt{a_1^2 + a_2^2}

This gives the straight-line distance from the origin to the tip of the vector.

The Dot Product

The dot product combines two vectors into a single number that reflects how aligned they are:

a⃗⋅b⃗=a1b1+a2b2\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2
  • If the result is positive: the vectors point in a similar direction;
  • If the result is zero: the vectors are perpendicular;
  • If the result is negative: they point in opposite directions.

Example

If a⃗=(1,2)  and  b⃗=(3,4) \vec{a} = (1, 2)\ \ \text{and}\ \ \vec{b} = (3, 4), then:

a⃗⋅b⃗=1⋅3+2⋅4=11\vec{a} \cdot \vec{b} = 1 \cdot 3 + 2 \cdot 4 = 11
question mark

If a⃗=(1,0), b⃗=(0,1)\vec{a} = (1, 0),\ \vec{b} = (0, 1). Then their dot product is:

Select the correct answer

Everything was clear?

How can we improve it?

Thanks for your feedback!

SectionΒ 4. ChapterΒ 1
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