1Let u v w and x be vectors in R2 Suppose that u is a lin

1.Let u , v , w , and x be vectors in R2. Suppose that u is a linear combination of v and w and that v is a linear combination of w and x . Show that u is a linear combination of w and x .

2.Let x , y , and w be vectors in R2. Suppose that x and y are both orthogonal to w . Verify that 2x y is orthogonal to w .

Solution

1)Solution for the first question

As u is a linear combination of v & w, we can express u as below

u = a1.v + b1.w

Similarly, v is a linear combination of w and x , we can express v as below

v = a2.w + b2.x

where a1, a2, b1, b2 are constants

Substituting for v in u , we get

u = a1.( a2.w + b2.x)   + b1.w

u = (a1.a2+b1).w + a1.b2.x

Hence u is a linear combination of w and x , hence proved

2) Solution for second question

Let          x = [a1, b1] &

                y = [a2, b2]

                w = [a3,b3]

As x and y are orthogonal to w, their dot products is zero, hence

a1.a3 + b1.b3 = 0        -   {1}

a2.a3 + b2.b3 = 0             {2}

Let us now compute the scalar product of [2x -y] & w.

[2x -y] can be represented as [2a1-a2, 2b1-b2]

Scalar product for the two vectors [2x -y] & w and is

{2a1-a2).a3 + (2b1-b2).b3

=2(a1.a3 + b1.b3) – (a2.a3+b2.b3)

= 0      {From 1 and 2 above}

Two vectors are orthogonal if and only if their scalar products are zero, hence proved that 2x -y is orthogonal to w