**Linear combination:** Let V be a vector space over a field K. A vector v in V is a linear combination of vectors u_{1}, u_{2},—–, u_{m} in V if there exist scalars a_{1}, a_{2}, —–, a_{m} in K such that

v= a_{1}u_{1}+ a_{2}u_{2}+—+ a_{m}u_{m}

**Example:** (Linear combination in R^{n}): Suppose we want to express v= (3, 7, -4) in R^{3} as a linear combination of the vectors

u_{1}=(1, 2, 3) u_{2}= (2, 3, 7) u_{3}= (3, 5, 6)

We suppose scalars x, y, z such that

v=xu_{1}+ yu_{2}+ zu_{3}; that is

=x+y+z

⇒x+2y+3z=3

2x+3y+5z=3

3x+7y+6z=-4

Reducing the system to echelon form yields

x+2y+3z=3

-y -z =1

Y -3z =-13

Again, reducing the system to echelon form yields

x+2y+3z=3

-y -z =1

-4z =-12

From last equation we get, z=3

From second equation we get, y = -4

And from first equation we get, x=2

Thus, v= v=2u_{1}-4u_{2}+3u_{3}

**Spanning set:** Let V be a vector space over K. Vectors u_{1}, u_{2}, …, u_{m} in V are said to span V or to form a spanning set of V if every v in V is a linear combination of the vectors u_{1}, u_{2}, …, u_{m} that is, if their exist scalars a_{1}, a_{2}, …, a_{m} in K such that

v= a_{1}u_{1}+ a_{2}u_{2}+—+ a_{m}u_{m}

**Subspace:** Let V be a vector space over a field K and let W be a subset of V. Then W is a subspace of V if W is itself a vector space Over K with respect to the operations of vector addition and scalar multiplication on V.