Linear Algebra
Practice Midterm 2
1. Consider A=
ļ£®
ļ£Æ
ļ£Æ
ļ£Æ
ļ£Æ
ļ£Æ
ļ£°
1ā1 1
2ā2 2
0 1 ā3
2 1 ā7
1 0 ā2
ļ£¹
ļ£ŗ
ļ£ŗ
ļ£ŗ
ļ£ŗ
ļ£ŗ
ļ£»
.
(a) Find the echelon form of A.
(b) Find the dimension of the four fundamental subspaces of A.
(c) Find a basis for each of the four fundamental subspaces of A.
(d) Use your bases to show explicitly that the row space of Ais orthogonal to the null space of
A.
2. Consider B=ļ£®
ļ£Æ
ļ£°
1ā1
2ā2
0 1
ļ£¹
ļ£ŗ
ļ£».
(a) Use Gram-Schmidt on the columns of Bto produce an orthornomal set.
(b) Find the QR factorization of B.
(c) The column space of Bis a subspace of what space?
(d) Find the matrix which projects onto the column space of B.
(e) Find the projection of the vector a=ļ£®
ļ£Æ
ļ£°
1
1
1
ļ£¹
ļ£ŗ
ļ£»onto the column space of B.
(f) Is the system Bx=aconsistent? Why or why not?
(g) Find the projection of the vector b=ļ£®
ļ£Æ
ļ£°
1
2
1
ļ£¹
ļ£ŗ
ļ£»onto the column space of B.
(h) Is the system Bx=bconsistent? Why or why not?
3. A linear transformation in R3stretches the x-axis by a factor of 3, shrinks the y-axis by a factor
of 2, and projects onto the xy plane.
(a) Find the matrix of this linear transformation.
(b) Describe the column space and null space of this transformation.
(c) Is this transformation invertible?
4. (a) What is the rank of the nĆnmatrix with every entry equal to 1?
(b) What is the rank of the nĆncheckerboard matrix with aij = 0 for i+jeven and aij = 1
for i+jodd?
5. Miscellaneaous Proofs.
(a) If Ais an nĆnmatrix of rank nand A2=A, prove that A=I.
(b) If you have a set of vectors v1,v2, . . . vnwhich are an orthonormal basis for Rn, prove that
v1vT
1+v2vT
2+Ā· Ā· Ā· vnvT
n=I.
(c) Prove that if a set of vectors v1,v2,. . . vnare orthonormal, they must be linearly independent.
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