extra_practice_s20-solutions

Math 1553: Some Additional Final Exam Practice Problems
Spring 2020

These problems are for extra practice for the final. They are not meant to be 100% compre-
hensive in scope.

1. Define the following terms: span, linear combination, linearly independent, linear transforma-
tion, column space, null space, transpose, inverse, dimension, rank, eigenvalue, eigenvector,
eigenspace, diagonalizable, orthogonal.

2. Let A be an m  n matrix.
a) How do you determine the pivot columns of A?
b) What do the pivot columns tell you about the equation Ax = b?
c) What space is equal to the span of the pivot columns?
d) What is the difference between solving Ax = b and Ax = 0? How are the two solution
sets related geometrically?
e) If rank(A) = r, where 0 ≤ r n, then how many columns have pivots? What is the
dimension of the null space?

Solution.
a) Do row operations until A is in a row echelon form. The leading entries of the rows are
the pivots.
b) If there is a pivot in every column, then Ax = b has zero or one solution. Otherwise,
Ax = b has zero or infinitely many solutions.
c) The pivot columns form a basis for the column space Col A.
d) Suppose that Ax = b has some solution x0. Then every other solution to Ax = b has the
form x0 + x , where x is a solution to Ax = 0. In other words, the solution set to Ax = b is
either empty, or it is a translate of the solution set to Ax = 0 (the null space).
e) If rank(A) = r then there are r pivot columns. The null space has dimension n − r.

,3. Let T : Rn → Rm be a linear transformation with matrix A.
a) How many rows and columns does A have?
b) If x is in Rn, then how do you find T (x )?
c) In terms of A, how do you know if T is one-to-one? onto?
d) What is the range of T ?

Solution.
a) A has m rows and n columns.
b) T (x ) = Ax .
c) T is one-to-one if and only if A has a pivot in every column. T is onto if and only if A has
a pivot in every row.
d) Col A.

4. Let A be an invertible n  n matrix.
a) What can you say about the columns of A?
b) What are rank(A) and dim Nul A?
c) What do you know about det(A)?
d) How many solutions are there to Ax = b? What are they?
e) What is Nul A?
f) Do you know anything about the eigenvalues of A?
g) Do you know whether or not A is diagonalizable?

Solution.
a) The columns are linearly independent. They also span Rn.
b) rank(A) = n and dim Nul A = 0.
c) det(A) /= 0.
d) The only solution is x = A−1 b.
e) Nul A = 0.
f) They are nonzero.
g) No, invertibility has nothing to do with diagonalizability.




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, 5. Let A be an n  n matrix with characteristic polynomial f (λ) = det(A λI ). (note: your
instructor may have defined the characteristic polynomial as det(λI−A). In either case, A will
have the same eigenvalues and eigenvectors)
a) What is the degree of f (λ)?
b) Counting multiplicities, how many (real and complex) eigenvalues does A have?
c) If f (0) = 0, what does this tell you about A?
d) How can you know if A is diagonalizable?
e) If n = 3 and A has a complex eigenvalue, how many real roots does f (λ) have?
f) Suppose f (c) = 0 for some real number c. How do you find the vectors x for which
Ax = cx ?
g) In general, do the roots of f (λ) change when A is row reduced? Why or why not?

Solution.
a) n
b) n
c) A is not invertible, since 0 is an eigenvalue.
d) If f has n distinct roots, then A is diagonalizable. Otherwise, you have to check if the
dimension of each eigenspace is equal to the algebraic multiplicity of the corresponding
eigenvalue.
e) Complex roots come in pairs, so f has one real root.
f) You compute Nul(A − c I ). (note: this is the same as Nul(c I A))

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g) Yes, row reduction does not preserve eigenvalues. For instance, is row equivalent to




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