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Module - 1 Linear Algebra and Vector Algebra

Lesson - 2 Vectors Algebra

**Overview**

Vectors and vector spaces are principal concepts in arithmetic that have a wide extend of applications in material science, designing, computer science, and numerous other areas. A vector could be a mathematical object that has both magnitude (length) and direction, and it can be spoken to geometrically as an arrow in space. A vector space may be a collection of vectors that fulfills certain axioms, such as closure beneath addition and scalar multiplication.

**Scalar and Vectors**

Scalars and vectors are two crucial concepts in arithmetic, especially within the field of linear algebra.

- A scalar could be a single numerical amount, such as a genuine number or complex number. Scalars are frequently utilized to represent physical amounts that have size but no direction, such as temperature, mass, or time. Scalars can be added, subtracted, increased, and divided utilizing the usual arithmetic operations.
- A vector, on the other hand, may be a mathematical object that has both magnitude and direction. Vectors are frequently utilized to represent physical quantities that have both measure and direction, such as velocity, force, or displacement. Vectors can be spoken to graphically as arrows, where the length of the arrow represents the magnitude of the vector and the heading of the arrow represents the direction of the vector.

In mathematics, vectors are often represented using coordinates or components.

For outline,** a two-dimensional vector inside the xy-plane can be represented as (x, y), where x and y are the x- and y-components of the vector, independently. In three-dimensional space, a vector can be spoken to utilizing three coordinates, and so on.

In expansion to the usual arithmetic operations, vectors can be included and subtracted utilizing vector addition and subtraction, respectively. Vector multiplication is additionally characterized, and there are a few diverse sorts of vector multiplication, including dot product, cross product, and scalar triple product.

Scalars and Vectors are critical concepts in arithmetic that are utilized in numerous distinctive zones, counting material science, designing, and computer science.

**Vector Geometry**

**Vector Magnitude and Norm:**

The magnitude (or length) of a vector v = (x, y) is given by the formula

||v|| = sqrt(x2 + y2)

which represents the distance from the origin to the point (x, y).

For example, the magnitude of the vector

v = (3, 4)

is given by,

||v|| = sqrt(32 + 42) = 5.

A vector's norm is the vector's magnitude divided by its length, denoted by ||v||/|v|.

**Vector Addition:**

Given two vectors u and v in a vector space, their sum is another vector denoted by u+v, where the i-th component of u+v is given by

(u+v)_i = u_i + v_i

for i=1, 2,..., n.

**Example:**

Let u = (1, 2, 3) and v = (4, 5, 6). Then,

u+v = (1+4, 2+5, 3+6) = (5, 7, 9)

**Properties**:

**Commutativity**: u+v = v+u

**Associativity**: (u+v)+w = u+(v+w)

**Identity element**: There exists a vector 0 such that u+0 = u for all vectors u

**Inverse element:** For every vector u, there exists a vector -u such that u+(-u) = 0

**Vector Subtraction:**

Given two vectors u and v in a vector space, their difference is another vector denoted by u-v, where the i-th component of u-v is given by

(u-v)_i = u_i - v_i

for i=1, 2,..., n.

**Example**:

Let u = (1, 2, 3) and v = (4, 5, 6). Then,

u-v = (1-4, 2-5, 3-6) = (-3, -3, -3).

**Properties:**

u-v = u+(-v)

Subtraction is not commutative, i.e., u-v is not equal to v-u in general.

**Scalar multiplication:**

Given a vector u in a vector space and a scalar c, their product is another vector denoted by cu, where the i-th component of cu is given by

(cu)_i = c*u_i for i=1,2,...,n.

**Example**: Let u = (1, 2, 3) and c = 2. Then,

cu = (21, 22, 2*3) = (2, 4, 6).

**Properties**:

**Distributivity**: c(u+v) = cu + cv

**Associativity**: (c1c2)u = c1(c2u)

**Identity element**: 1u = u for all vectors u

**Zero element**: 0u = 0 for all vectors u

**Dot product:**

Given two vectors u and v in a vector space, their dot product (or inner product) is a scalar denoted by

u.v or <u,v> where

u.v = u1v1 + u2v2 + ... + unvn.

**Example**: Let u = (1, 2, 3) and v = (4, 5, 6). Then,

u.v = 14 + 25 + 3*6 = 32.

**Properties**:

**Commutativity**: u.v = v.u

**Distributivity**: u.(v+w) = u.v + u.w

**Associativity**: c(u.v) = (cu).v = u.(cv)

**Orthogonality**: u.v = 0 if and only if u and v are orthogonal (i.e., perpendicular) to each other

**Cross product:**

Given two vectors u and v in a three-dimensional vector space, their cross product is another vector denoted by u x v, where the i-th component of u x v is given by:

u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)

Example: Let u = (1, 2, 3) and v = (4, 5, 6). To find u x v, we can use the following formula:

u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)

Plugging in the coordinates of u and v, we get:

$$ u x v = (26 - 35, 34 - 16, 15 - 24) = (-3, 6, -3) $$

Therefore, u x v = (-3, 6, -3).

**Properties**:

**Anticommutativity**: v x u = -u x v for any vectors u and v.**Distributivity**: u x (v + w) = u x v + u x w for any vectors u, v, and w.**Scalar multiplication**: (cu) x v = c(u x v) = u x (cv) for any scalar c and vectors u and v.**Magnitude**: The magnitude of u x v is equal to the area of the parallelogram formed by u and v.**Orthogonality**: u x v is orthogonal (i.e., perpendicular) to both u and v.**Non-associativity:**(u x v) x w is not equal to u x (v x w) in general.**Parallelism**: u x v = 0 if and only if u and v are parallel (i.e., one is a scalar multiple of the other).

**Linear Independence and Spanning**

Linear independence and spanning are two basic concepts in linear algebra that are utilized to portray the properties of a set of vectors in a vector space.

A set of vectors in a vector space is said to be linearly independent within the occasion that none of the vectors inside the set can be communicated as a direct combination of the other vectors inside the set. More formally, a set of vectors {v1, v2, ..., vn} is straightly independent in the event that the condition

$$ a1v1 + a2v2 + ... + anvn = $$

On the other hand, a set of vectors in a vector space is said to span the space in case each vector interior the space can be communicated as a straight combination of the vectors interior the set. More formally, a set of vectors {v1, v2, ..., vn} ranges the vector space on the off chance that for each vector v interior the space, there exist scalars a1, a2, ..., an such that

$$ v = a1v1 + a2v2 + ... + anvn $$

The set of vectors {v1, v2, ..., vn} is said to make a premise for the vector space in case it is both linearly independent and ranges the space. A basis for a vector space is critical since it permits us to uniquely represent any vector within the space as a straight combination of the premise vectors.

**Example**:

Let's consider the following set of equations:

$$ 2x + 3y = 5

$$

$$ 4x + 6y = 10

$$

To determine if these equations are linearly independent, we need to see if one equation can be expressed as a linear combination of the other. We can do this by manipulating the equations to isolate one variable and see if we can use that to solve for the other variable in both equations.

Starting with the first equation, we can isolate x by subtracting 3y from both sides:

$$ 2x = 5 - 3y $$

Now we can solve for x:

$$ x = (5 - 3y)/2 $$

Next, we substitute this expression for x into the second equation:

$$ 4((5 - 3y)/2) + 6y = 10 $$

Simplifying this equation gives:

$$ 10 - 6y + 6y = 10 $$

We can see that this equation simplifies to 10 = 10, which is always true. This means that we can express one equation as a linear combination of the other, and therefore these equations are linearly dependent.

**Vector Projections**

Vector projection is a process of finding the component of one vector onto another. Given two non-zero vectors, u and v, the projection of u onto v is a scalar value that represents the length of the component of u that lies in the direction of v.

To find the projection of a vector "a" onto another vector "b", we can use the following formula:

proj_b(a) = (a . b / ||b||^2) * b

Here, "a . b" represents the dot product of vectors "a" and "b", "||b||" represents the magnitude of vector "b", and "*" represents scalar multiplication.

For example, suppose we have two vectors:

a = (2, 4, 1)

b = (1, -1, 2)

To find the projection of "a" onto "b", we first need to calculate the dot product of "a" and "b":

a . b = (2 * 1) + (4 * -1) + (1 * 2) = 0

Next, we need to calculate the magnitude of vector "b":

||b|| = sqrt(1^2 + (-1)^2 + 2^2) = sqrt(6)

Finally, we can use the formula to calculate the projection of "a" onto "b":

proj_b(a) = (a . b / ||b||^2) * b

= (0 / 6) * (1, -1, 2)

= (0, 0, 0)

This means that the projection of "a" onto "b" is the zero vector, indicating that "a" is perpendicular to "b".

**Orthogonality**

Two vectors are considered orthogonal if they are perpendicular to each other, i.e., their dot product is zero. Geometrically, this means that the angle between the two vectors is 90 degrees.

- Here is an example of two orthogonal vectors:

v = (1, 0, 0)

w = (0, 1, 0)

To check if these vectors are orthogonal, we can compute their dot product:

v · w = (1)(0) + (0)(1) + (0)(0) = 0

Since the dot product is zero, we can say that v and w are orthogonal.

Note that the dot product is defined as the sum of the products of the corresponding components of two vectors. In other words, for two vectors u and v of the same dimension, their dot product is given by:

u · v = u1v1 + u2v2 + ... + unvn

where u1, u2, ..., un are the components of u and v1, v2, ..., vn are the components of v.

- If the dot product of two vectors is not zero, then they are not orthogonal. For example, consider the following two vectors:

x = (1, 2, 3)

y = (4, 5, 6)

Their dot product is:

x · y = (1)(4) + (2)(5) + (3)(6) = 32

Since the dot product is not zero, we can say that x and y are not orthogonal.

**Applications of Vectors Algebra**

- Machine Learning and Information Investigation: Vectors and vector spaces are too utilized in machine learning and information examination to speak to information focuses and highlights. For illustration, in a text classification assignment, each record can be represented as a vector in a high-dimensional space where each measurement compares to a word within the vocabulary. The similitude between two reports can be measured utilizing the dot product of their comparing vectors.
- Material science and Designing Applications Vectors and vector spaces are utilized broadly in material science and designing to portray amounts such as drive, speed, and speeding up. For illustration, the movement of an object can be depicted by a vector in three-dimensional space that speaks to its position, speed, and increasing speed.

**Key takeaways**

- Vectors and vector spaces are principal concepts in science utilized in material science, building, computer science, and numerous other areas.
- Scalars are numerical quantities utilized to speak to physical amounts with magnitude but no direction, whereas vectors are scientific objects representing physical amounts with both size and direction, such as speed, drive, or displacement.
- Vectors can be represented graphically as arrows with length representing the magnitude of the vector and direction representing its direction.
- Vectors are often represented using coordinates or components.
- Vector addition, subtraction, and multiplication are defined, with different types of vector multiplication, such as dot product, cross product, and scalar triple product.
- Vector geometry involves concepts such as vector magnitude and norm, vector addition, vector subtraction, and their properties.

**Conclusion**

In conclusion, vectors and vector spaces are principal concepts in science that are broadly utilized in material science, designing, computer science, and other areas. Scalars and vectors are two principal concepts in science, especially in linear algebra, where a scalar may be a single numerical amount, and a vector could be a mathematical object that has both magnitude and direction. By and large, vectors and vector spaces are basic concepts that play a critical part in different areas of study.

**Quiz**

**What is a vector?** a. A mathematical object that has only magnitude b. A mathematical object that has only direction c. A mathematical object that has both magnitude and direction d. A mathematical object that has neither magnitude nor direction

**Answer**: c. A mathematical object that has both magnitude and direction

**How is a vector represented geometrically?** a. As a straight line b. As a circle c. As an arrow in space d. As a point in space

**Answer**: c. As an arrow in space

**What is the formula for calculating the magnitude of a vector?** a. ||v|| = sqrt(x^2 - y^2) b. ||v|| = sqrt(x^2 + y^2) c. ||v|| = x + y d. ||v|| = x - y

**Answer**: b. ||v|| = sqrt(x^2 + y^2)

**What is the i-th component of the sum of two vectors u and v?** a. u_i - v_i b. u_i / v_i c. u_i + v_i d. u_i * v_i

**Answer**: c. u_i + v_i

**What is the identity element of vector addition?** a. There is no identity element for vector addition b. A vector that is the additive inverse of itself c. A vector that is the multiplicative inverse of itself d. A vector that leaves other vectors unchanged when added

**Answer**: d. A vector that leaves other vectors unchanged when added

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