Download Worksheet 9 - Midterm 1 Review Math 54, GSI

Worksheet 9 - Midterm 1 Review
Math 54, GSI: Andrew Hanlon
1
Straightforward problems
1. Is the set of vectors below linearly independent? Do these vectors span all of R4 ?
    

1
1
10




    

0
2
6
 , ,

−2 0 −14





0
3
9
2. For the matrix


1 −1 0 2
A = 2 4 8 0 ,
0 3 9 1
find a basis for Null(A) and a basis for Col(A).
3. Consider the two matrices below.


1 2
A = 4 3
1 5
1 5 10
B=
2 4 8
Compute AB and BA.
4. Compute the determinant of the matrix below.


2
4
5 1
1
0 −1 1

A=
−2 6 −3 3
2 −2 4 4
Is A invertible?
5. Solve the linear system


 
2 3
5
1
1 0 −1 x = 0
1 −2 1
0
using all three methods that we know (row reduction, finding the inverse matrix, and Cramer’s
rule).
2
Intermediate problems
6. Compute the standard matrix of the linear map R2 → R2 that is defined by first rotating
counterclockwise by 3π/2 then sending (1, 0) to (1, 1) while fixing (0, 1).
7. Show that there is no 2 × 4 matrix A that has

 
x1




 

x
2
4


Null(A) =   ∈ R : x1 = x2 = x3 = 0
x3






x4
8. Given that det(A) = 2, det(B) = 5, C is obtained by switching two rows of A and multiplying
a column by 3, and all three matrices are 3 × 3. Compute the following or say if there is
not enough information to uniquely determine the answer: det(AB), det(2A), det(A + B), and
det(C).
9. True or false: The subset of n × n matrices consisting of matrices with determinant 1 is a
subspace. Justify your answer.
10. True or false: Let Ao be a fixed n × n matrix. Then, the subset of n × n matrices consisting of
matrices B that commute with Ao ( this means Ao B = BAo ) is a subspace of Mn×n . Justify
your answer.
11. True or false: The determinant of AT A is never negative. Justify your answer.
3
Challenging problems
12. True or false: AT A = AAT for every n × n matrix A. Justify your answer.
13. True or false: Every subspace U ⊂ Rn is the null space (same as kernel) of a linear transformation T : Rn → Rk for some k. Justify your answer.
14. An n × n matrix is said to be symmetric if AT = A and anti-symmetric if AT = −A. Show
that the sets of symmetric and anti-symmetric matrices are subspaces of Mn×n . Compute their
dimensions and show that the span of vectors in their union is all of Mn×n .
15. Consider
R 1the map I : C(R) → R where C(R) is the space of continuous functions on R given by
I(f ) = 0 f (t) dt. Check that I is linear. Is I injective? Is I surjective? Find the dimensions
of the kernel and range of I when it is restricted to the supspace of polynomials of degree ≤ n,
Pn , of C(R).
16. For a vector space V , a linear transformation P : V → V is said to be a projection if P ◦ P = P .
Show that P is the identity map on its range. Give an example of a projection.