math test
MiracleFY
Math270Test3ReviewSpring2020.pdf
MATH 270 TEST 3 REVIEW
1. Given subspaces H and K of a vector space V , the sum of H and K, written as H +K,
is the set of all vectors in V that can be written as the sum of two vectors, one in H
and the other in K; that is
H + K = {w : w = u + v}, ∃u ∈ H and ∃v ∈ K. Show that H + K is a subspace of V .
2. Based on problem 1, show that H is a subspace of H +K and K is a subspace of H +K.
3. Find an explicit description of Nul A by listing vectors that span the null space for the
following matrix :
A =
1 −2 0 4 00 0 1 −9 0
0 0 0 0 1
4. Let A =
[ −6 12 −3 6
] and w =
[ 2
1
] .
Determine if w ∈ Col A. Is w ∈ Nul A ?
5. Define T : P2 → R2 by T(p) =
[ p(0)
p(1)
] .
For instance, if p(t) = 3 + 5t + 7t2, then T(p) =
[ 3
15
] .
Show that T is a linear transformation. [Hint : For arbitrary polynomials p, q ∈ P2, compute T(p + q) and T(cp) ].
6. Find a basis for the space spanned by the given vectors v1,v2,v3,v4,v5.
1
0
−3 2
,
0
1
2
−3
,
−3 −4 1
6
,
1
−3 −8 7
,
2
1
−6 9
7. Let v1 =
4−3
7
, v2 =
19 −2
, v3 =
711
6
and H = Span{v1,v2,v3}. It can be verified that 4v1 + 5v2 − 3v3 = 0. Use this information to find a basis for H.
8. Find the coordinate vector [x]B of x relative to the given basis B = {b1,b2,b3}.
b1 =
1−1 −3
, b2 =
−34
9
, b3 =
2−2
4
, x =
8−9
6
9. Use an inverse matrix to find [x]B for the given x and B.
B =
{[ 3
−5
] ,
[ −4 6
]} , x =
[ 2
−6
]
10. The set B = {1 + t2, t + t2, 1 + 2t + t2} is a basis for P2. Find the coordinate vector of p(t) = 1 + 4t + 7t2 relative to B.
11. Use coordinate vectors to test the linear independence of the set of polynomials.
Explain your work.
1 + 2t3, 2 + t− 3t2,−t + 2t2 − t3
12. Find the dimension of Nul A and Col A for the matrix shown below.
A =
1 −6 9 0 −2 0 1 2 −4 5 0 0 0 5 1
0 0 0 0 0
13. Assume matrix A is row equivalent to B. Find bases for Col A, Row A and Nul A of
the matrices shown below.
A =
2 −3 6 2 5 −2 3 −3 −3 −4 4 −6 9 5 9 −2 3 3 −4 1
, B =
2 −3 6 2 5 0 0 3 −1 1 0 0 0 1 3
0 0 0 0 0
14. If a 3 × 8 matrix A has rank 3, find dim(Nul A), dim(Row A) and rank(AT ).
15. Let A = {a1,a2,a3} and B = {b1,b2,b3} be bases for a vector space V and suppose a1 = 4b1−b2, a2 = −b1 +b2 +b3 and a3 = b2−2b3. Find the change-of-coordinate matrix from A to B. Then find [x]B for x = 3a1 + 4a2 + a3.
16. Let B = {b1,b2} and C = {c1,c2} be bases for R2. Find the change-of-coordinate matrix from B to C and the change-of-coordinate matrix from C to B.
b1 =
[ 7
5
] , b2 =
[ −3 −1
] , c1 =
[ 1
−5
] , c2 =
[ −2 2
]
17. In P2, find the change-of-coordinate matrix from the basis B = {1 − 2t + t2, 3 − 5t + 4t2, 2t + 3t2} to the standard basis C = {1, t, t2}. Then find the B-coordinate vector for −1 + 2t.
18. Find a basis for the eigenspace corresponding to the eigenvalue.
A =
4 2 3−1 1 3
2 4 9
, λ = 3
19. Find the characteristic polynomial and the eigenvalues of the following matrix.[ 3 −2 1 −1
]
20. Find the characteristic polynomial of the following matrix. 6 −2 0−2 9 0
5 8 3
