Today we will be looking at the basic concepts and properties of linear algebra. Linear algebra can also be called as vector algebra, as it is the study of vector spaces. You may now wonder what exactly vectors are. Well, quantities are classified into two types such as scalars and vectors. A scalar quantity is a quantity which has only magnitude and no direction. A quantity which has both magnitude and direction is called as a vector.
Let us look at some definitions and properties of vectors which will make the study of the vector spaces much simpler.
Position Vector: The position vector locates any given point in the three dimensional rectangular coordinate system. Let us recollect that the three dimensional rectangular coordinate plane has x, y and z axes. The origin point for such coordinate system is given by (0,0,0). The position vector or any point P in space is given by OP (x, y, z). The magnitude of the vector |OP| can be given by √(x2 + y2 + z2).
So, we have |OP| = √(x2 + y2 + z2).
Let us take a look at one of such example question.
Question: What is the position vector of the point (3, 4, 5). Find the magnitude of the given position vector?
The position vector is found from origin (0, 0, 0) to the point P. Follow the steps in the diagram below for finding the magnitude of the given vector:
Let us now take a look at some basic types of vector spaces.
Null vector: A null vector is also called as the zero vector. Remember the starting point of the vector is called the initial point and the ending point is called as the terminal point. The vector for which both the initial point and terminal point are the same is called as the zero vector. Like the name suggests the magnitude of a zero vector is always equal to zero. The null vectors are written as PP, AA and so on.
Unit Vector: The magnitude of a vector when equal to 1 is called the unit vector.
Need an example? Here you go!
Find the magnitude of the vector AB given by (0, 1, 0).
Apply the formula for the magnitude of the vector. This is what you get:
Equal vector: Like the name suggests the equal vectors are the vectors which are the same in magnitude and direction. If AB and CD are two equal vectors, then they are given by AB = CD.
Negative Vector: Well again like the name suggests the vectors which have the same magnitude and are opposite in the direction are called the negative vectors. If OP is a vector than PO is called the negative vector. This can be represented as OP = – PO.
Here AB = – BA. So BA is the negative vector for vector AB.
Co-initial Vectors: Any given two vectors are called as co-initial vectors if both the given vectors have the same initial point. For example OA and OB are two co-initial vectors.
Collinear Vectors: Any two given vectors are called collinear vectors when both vectors are parallel to the same line.
In the above diagram AB and CD are parallel to the same line, hence the two vectors are collinear to each other.
Addition of vectors: Most students find it quite confusing to understand the addition of vectors. In simple terms here is how you can follow it:
In the above diagram E is the initial point for the vector EF and terminal point for vector DF. So to obtain the resultant we start from the initial point of one vector (DE) to the terminal point of another vector (EF). Hence the resultant of the vectors is DF.
This can be expressed as DF (resultant vector) = DE + EF.
Now that you are aware of the vector additions let us look at the properties of the vector space addition.
Commutative Property: Vector addition is commutative. Consider two vectors a and b, they can be expressed as: Associative property: Vector addition is also associative. Like associative property for numbers we need 3 vectors to express the associative property for vectors. The property can be written as follows: Scalar Multiplication: Now let us take a quick look at multiplying a vector by a scalar quantity. Let there be a scalar number or a constant k. Take a vector a. Now we can obtain a new vector by multiply with the constant k. This is expressed as ka, always remember that the vector obtained after the scalar multiplication is collinear to the given vector. So ka is collinear to vector a. The vector has positive of negative direction depending on the sign of k. Here are some examples to give better understanding: In the above diagram there is vector which is then multiplied with a scalar 2. Here 2 is a positive number so the direction of the vector 2a is the same but the magnitude of vector is twice the initial vector a. In the above diagram the vector is multiplied by the scalar number -3. Since the scalar is a negative number, the direction of the vector is the reverse to the direction of vector a. The magnitude of the vector will be thrice the magnitude of vector a. Do you now feel you have a better idea of the basics of the vector spaces? Well guess what this is just the beginning and there is lot more to learn and discover in linear algebra. So do not forget, keep practicing!
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