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MAT 208 Test #2
1.) Find a basis for the a.) column space and b.)null space for the matrix A below. Then provide the nullity and rank
of the matrix.
2.) Consider the following vectors.
𝒖 = (5, −1, 2) and 𝒗 = (2, −1, −3)
Compute the following:
a.) |𝒗|
b.) |𝒖|
c.) 𝒖 ∙ 𝒗
d.) 𝑝𝑟𝑜𝑗 𝑣
e.) The angle between the vectors.
3.) Find an orthonormal basis for R3 spanned by the vector space 𝑆 = {(1, 2, 2), (−1, 0, 2), (0,0,1)}.
1 2 0 
4.) a.) Find an orthogonal basis for  using the column space of D=  0 2 3 


1 0 2 
3
b.) Make an orthonormal basis from the columns of D.
c.) Find the vector [ 1 2 3]T as a linear combination of the orthonormal basis above.
5.) Given a = (1,2,3) b=(0,1,2)
a. Find the projections projba
1 0 
b. Find the projection of b onto the column space of A =  3 2  Write the projection as a linear combination


 0 1 
of the columns of A.
c.
Use Gram-Schmidt process to generate an orthonormal set of vectors that span the column vectors of A.