Addition and multiplication with a scalar

A scalar is a real number used in the context of vectors or matrices, which defines vector space. The multiple scalars that tell us the quantity, such as direction and magnitude, is called a vector.

In linear algebra, real numbers or elements are called scalars which relate to vector space through the operation of scalar multiplication where one vector is multiplied by a scalar to produce another vector. Vector space is associated with complex numbers, their scalars are also associated with complex numbers. In other words, Scalar product = Multiplication of two vectors.

ADDITION & MULTIPLICATION 

• The addition and scalar multiplication of the vectors can be done using algebraic or geometric numbers. 

• A unit vector is a vector of length 1 that is parallel to one of the axes. In the 2D plane, the unit vectors are often symbolised as i and j.

• Two vectors are said to be equal if their directions are the same.

• A zero vector is a vector that has 0 magnitudes and an arbitrary direction.

• The negative vector is a vector in which the magnitude is the same as that of a vector but the direction is opposite to that of a vector.

MULTIPLICATION OF MATRICES

When a vector is multiplied by a scalar quantity, the magnitude of the vector changes with the magnitude of the scalar while the direction of the vector remains fixed.

Let a matrix A & multiply it by any scalar k∈R.

By multiplying all the entries of v(order 1×2) by k.

Let v = 〈 v 1 , v 2 〉 

kv = k 〈 v 1 , v 2 〉 = 〈k v 1 ,k v 2 〉

PROPERTIES OF SCALAR MULTIPLICATION OF MATRIX

If P = [pij] and Q = [qij] are two matrices of same order p × q, and k and l are scalars, then:

• k(P + Q) = kP + kQ,
  (k + l)P = kP + lP

• k (P + Q) = k ([pij] + [qij])
  = k[pij + qij] = [k(pij + qij)] = [(kpij) + (kqij)]
  = [kpij] + [kqij] = k[pij] + k[qij] = kP+ kQ

• (k + l)P = (k + l)[pij]
  = [(k + l)pij] + [kpij] + [lpij] = k[pij] + l[pij] = kP + lP.

GRAPHICALLY

The vector changes its length, while its direction remains the same. If the vector’s magnitude was |v|, after getting multiplied by a scalar, it will become |av| = a|v|.

•  If |a| > 1 new vector is longer.

• If |a| < 1 new vector is shorter. 

• If a < 0, the new vector points in the opposite direction than the original one.

ADDITION OF VECTORS

For drawing the vectors geometrically, draw the vectors to a common scale and then place them according to head. The vector that connects the tail of the first vector to the head of the second vector is the sum. 

Points to be kept in mind while adding two vectors:

1. Vectors behave independently of each other.

2. Vector addition is finding the resultant of a number of vectors acting upon a body.

3. Commutative law is applied in vector addition which means the resultant vector is independent of the order of vectors.

4. Only matrices with the same order can be added to each other.

5. Vectors can be added geometrically only.

GRAPHICALLY

To add 2 vectors, place the initial point of the second vector (do not change the length or direction) on the terminal of the first. After that, join the initial point of the first vector to the terminus of the second vector. The joining line so obtained, will represent the sum of the 2 vectors.

PROPERTIES OF ADDITION IN A MATRIX

• Commutative property i.e, A + B = B+ A

• Associative Property i.e, A+ (B + C) = (A + B) + C

• Additive identity property i.e A+O = A

• Additive inverse property i.e A+ (-A) = O.

• Closure Property i.e A + B = C.

CONCLUSION

In this article, we have learned about the addition of vectors and multiplication of scalar, their properties, the concept of vector and scalar, graphical ways of solving, illustrations of examples with different orders of the matrix. We have also learned how to add and multiply the matrices using vectors and scalars.