Download Sample Problems for Midterm 2 1 True or False: 1.1 If V is a vector

Sample Problems for Midterm 2
1
True or False:
1.2
If V is a vector space, then (−1) u = −u for any vector u in V.
The set of all matrices c db where  ≥ 0 is a subspace of M2×2 .
1.3
If S = { v1 , v2 , v3 } is a spanning set, then T = { v1 , v3 } is also a spanning set.
1.4
If v 6= 0, then the set {v} is linearly independent.
1.5
If v3 is a linear combination of v1 and v2 , then spn{v1 , v2 , v3 } = spn{v1 , v2 }.
1.6
Every linearly independent set can be extended to a basis.
1.7
Every spanning set can be extended to a basis.
1.8
If {v1 , . . . , vn } is a spanning set and {w1 , . . . , wm } is linearly independent, then n ≤ m.
1.9
The matrices [1, 2, 3], [1, 0, 1], [1, 2, 3] are linearly independent.
1.1
1.10 The polynomials t, t 2 , t 3 span P3 .
1.11 A minimal spanning set is a basis.
1.12 A minimal independent set is a basis.
1.13 If dim V = n, then any independent set in V has at least n elements.
1.14 If dim V = n, then any independent set in V with n elements is a basis for V.
1.15 If W is a subspace of V, then dim W ≤ dim V.
1.16 If W = spn S and S is linearly independent, then S is a basis for W.
1.17 [v − w] S = [v] S − [w] S , for all vectors v, w, and all bases S.
1.18 Any transition matrix PS←T is non-singular.
1.19 Given three bases S1 , S2 , and S3 , we have that PS2 ←S1 · PS3 ←S2 = PS3 ←S1 .
1.20 If two matrices are row-equivalent, they they have the same row space.
1.21 If two matrices are column-equivalent, they they have the same row rank.
1.22 A linear system Ax = b, where A is n × n, has a unique solution if rnk A = n.
1.23 If rnk(A) = 0, then A is singular.
1.24 If nllity(A) = 0, then A is singular.
1.25 If an m × n matrix has rank n, then its columns are independent.
1.26 An n × n matrix has rank n if and only if its columns are a basis.
2
Give a proof of the following statements:
2.1
The set of all 1 × 3 matrices of the form [, b,  − 2b] is a subspace of R3 .
2.2
2.3
The set of polynomials of the form  t 4 −  t is a subspace of P5 .
The set of 2-vectors of the form 1 is not a subspace of R2 .
2.4
Let D be the disk in R2 centered at the origin with radius 1. Then D is not a subspace of R2 .
3
Consider the matrix:


1
1
4
1
2

0

A=
0

1
1
2
1
0
0
1
−1
0
0
1
6
0

1

2


2
1
2
3.1
Compute a basis for the row space of A.
3.2
Compute a basis for the column space of A.
3.3
Compute a basis for the kernel of A.
3.4
Compute rnk(A).
3.5
Compute nllity(A).
3.6
Is A non-singular?
3.7
How many solutions does Ax = b have?
4
Consider the following collections of polynomials:
S={
t3 ,
t 2 + t,
2t 2 + 1,
t+1
T={
}
t3 + t2 ,
t,
t3
1,
4.1
Show that both S and T are bases for P3 .
4.2
Find the transition matrix PS←T from the T-basis to the S-basis.
4.3
Find the transition matrix PT←S from the S-basis to the T-basis.
4.4
Let p(t) = t 3 + t 2 + t + 1. Compute the coordinate vector [p(t)] T .
4.5
Using your answers to previous questions, compute the coordinate vector [p(t)] S .
5
Let W be the subspace of P3 consisting of all polynomials of the form  t 3 + b t 2 + c t + d, where
3 − b − 5d = 0 and 4 − c + d = 0.
5.1
Compute the dimension of W.
5.2
Compute a basis for W.
6
Consider the following set of vectors in R3 :
S=
¦
”
1
—
0 ,
1
”
0
2
—
3 ,
”
1
2
—
3 ,
”
3
6.1
Show that S is linearly dependent.
6.2
Express one of the vectors in S as a linear combination of the others.
6.3
Find a basis for spn(S).
6.4
Find a subset of S that is a basis for spn(S).
7
S=
–
1
0
0
−1
™
–
,
0
0
1
0
™
–
,
2
0
1
−2
7.2
Show that S does not span M2×2 .
Determine whether the matrix 11 11 belongs to spn(S).
7.3
Find a basis for spn(S).
7.4
Find a subset of S that is a basis for spn(S).
8
6
6
™
«
Consider the following set of matrices in M2×2 .
¨
7.1
}
™
–
,
0
1
0
0
t+1
Consider the following set of polynomials in P3 :
S=
t 3 + t 2 + t + 1,
t 2 + t + 1,
8.1
Show that S does not span P3 .
8.2
Find a polynomial in P3 that does not belong to Spn(S).
8.3
Find a basis for P3 that contains S.
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