I have a two part question of which I have a semi-correct answer and can't figure out where I've gone wrong.
Let u=(-2,1) and v=(1,1)
i) Show u and v form an orthogonal basis for $\mathbb{R^2}$ wrt inner product
<u,v>=$2u_1v_1+u_1v_2+u_2v_1+5u_2v_2$. Use this basis to find an orthonormal basis by normalizing each vector.
First I showed u and v formed a basis of $\mathbb{R^2}$ by row reduction to get 2×2 identity matrix.
Second I showed u and v are orthogonal using the given inner product space and found that <u,v>=0
Third I normalised each vector to get ||u||=3 and ||v||=3 to give unit vectors for u to be a=(-2/3,1/3) and v to be b=(1/3,1/3).
ii) Express vector w=(1/3,4/3) as a linear combination of the orthonormal basis vectors obtained in part i.
To find linear combination of the two vectors I let w=$c_1$a+$c_2$b then solved to get $c_1=1$ and $c_2=3$ . These coefficients aren't correct because when I check and sub values in I get w=$c_1$a+$c_2$b=(-2/3,1)+3(1/3,1/3)=(1/3,2)
but the answer should be w=(1/3,4/3) after substituting.
I can't figure out where I've gone wrong. Any help greatly appreciated.
Best Answer
After you normalize u you should have $a = (-2/3, 1/3)$