We know that the space r2 is two-dimensional consider a

Question

We know that the space R² is two-dimensional. Mathematically, this means that we can generate any vector in R² using two base vector. For example, any vector in R² can be written as a weighted sum of i end j, is for same m, n ∈ R, v = m_i + n_j, where v is an arbitrary 2D vector. These base vectors are not unique. For this problem, we investigate whether or not a= (1, 1) and B = (1, -1) form base vectors,

(a)   Consider a vector v = (-3, 4). Find the vector projection of v onto a and b respectively, denote them by proj_av, and proj_bv.

(b)   For any vector v = (x, y) where x and y are some real numbers, find its vector projection on a and b, and show that

v = proj_av + proj_bv.

Show that R² can be also spanned by a and b, i.e. for any vector v we have v = ma+ nb for some real numbers m and n.