Let B1x and Cx1x be two order bases of P1 a By inspection fi

Let B=(1,x) and C=(x,1+x) be two order bases of P1.

a. By inspection, find the coordinate of vectors (p(x))_B and (p(x))_C using the vector p(x)=4-3x

b. Find the change of basis matrix (transition matrix) from B to C.

c. Use your answer from part b to compute the coordinate vector (p(x)_c.

d. Find the change of basis matrix (transition matrix) from C to B.

e. Use your answer from part d to compute the coordinate vector (p(x))_B.

Solution

[ 1    0 ]

3. The coordinate vector (p(x)_c is [ -1 1 ] [ 4 ] i.e. [ -7 ]

[ 1   0 ] [ -3 ] [ 4 ]

4. Since x = 0*1 + 1* x and since 1 +x = 1*1 + 1*x, hence the change of coordinate matrix from C to B

is [ 0   1 ]

[ 1   1 ]

5 The coordinate vector (p(x))_B is [ 0   1 ] [ -7 ] i.e. [ 4 ]

[ 1   1 ] [ 4 ] [-3 ]