Practice Problems Subspaces, Bases & Dimension Math 201-105-RE

1. Let u1 = (3, −1, 2) and u2 = (3, 1, 5). 8. Let u1 = (2, 3, −1, ), u2 = (5, 4, −1), u3 = (5, −3, 2),
(a) Express the vector v = (9, 11, 27) as a linear combination u4 = (0, 6, −2), u5 = (0, −15, 5).
of u1 and u2 if possible. Determine whether each set is linearly independent or linearly
dependent. In each case, state whether the span of the set is a
(b) Find k such that the vector w = (−5, 4, k) is a linear com- point, line, plane, or R3 .
bination of u1 and u2 . (a) {u1 , u2 }
     
6 9 7 (b) {u1 , u2 , u3 }
2. Let a1 = 3, a2 = −3, and b =  6 .
     
(c) {u2 , u3 , u4 }
4 5 h
(d) {u2 , u3 , u4 , u5 }
(a) Find h so that b is in Span{a1 , a2 }.
(e) {u4 , u5 }
(b) For the h that you found in the previous part, express b as a 9. Let u1 = (0, −5, 5, ), u2 = (0, 3, −3), u3 = (1, 1, 1),
linear combination of a1 and a2 . u4 = (1, 0, 1), u5 = (2, 2, 0), and 0 = (0, 0, 0).
Determine whether each set is linearly independent or linearly
3. Let b1 = (h, 5, 7), b2 = (−1, 3, 7), and b3 = (1, 1, 2).
dependent. (LI or LD?) In each case, state whether the span of
Find h so that b3 ∈ Span{b1 , b2 }.
the set is a point, line, plane, or R3 .
4. (a) Express the plane x − 3y + 4z = 0 as a span of vectors. (a) {u1 }

(b) Express the intersection of the two planes x − 3y + 4z = 0 (b) {u1 , u2 }

and 2x + z = 0 as a span of vectors. (c) {u1 , u2 , u3 }

5. Find an equation of the plane in form Ax + By + Cz = D (d) {u2 , u3 , u4 }

that is spanned by the vectors (2, 3, −1) and (4, 1, 5). (e) {u2 , u3 , u4 , u5 }

6. Let u1 = (2, 0, 3, −1), u2 = (−4, 0, −6, 2), u3 = (5, 5, 0, 3), (f) {u4 , u5 }
u4 = (1, 3, −6, 5), 0 = (0, 0, 0, 0). (g) {u4 , u5 , 0}
Determine whether each set is linearly independent or linearly
(h) {0}
dependent.
10. Determine if the following sets are subspaces. For those that are,
(a) {u1 }
express the set as a span of vectors. For those that are not, pro-
(b) {u1 , u2 } vide a counter-example to show it is not closed under VA or SM.
(a) S = {(x, y, z) ∈ R3 | x = 4s − t, y = s + 3t, z = 6s}
(c) {u1 , u2 , u3 }
(b) S = {(x, y, z) ∈ R3 | 3x + 4y − z = 2}
(d) {u2 , u3 , u4 }
(c) S = {(x, y, z) ∈ R3 | z 2 = xy}
(e) {u3 , u4 } (d) S = {(x, y, z) ∈ R3 | x + 2y − 3z = 0}
(f) {u3 , u4 , 0} (e) S = {(x, y, z) ∈ R3 | y ≥ x}

7. Let u1 = (5, 2, −1, 6), u2 = (3, 1, 0, 2), (f) S = {(x, y) ∈ R2 | x = 4 + 2t, y = −6 − 3t}
u3 = (1, 1, −2, 6), u4 = (1, 1, −2, 1), u5 = (1, 0, 0, 0). (g) S = {(x, y, z) ∈ R3 | y + z ≥ −1}
Determine whether each set is linearly independent or linearly (h) S = {(x, y, z) ∈ R3 | z = 2x − 3y}
dependent.   
 x
 

(a) {u1 , u2 } (i) S = y  ∈ R
  3
xy + z = 0
 
z
 
(b) {u1 , u2 , u3 }
  
 x
 
(c) {u1 , u2 , u3 , u4 }

(j) S = y  ∈ R3 2x = y − z
 
 
z
 
(d) {u2 , u3 , u4 , u5 }
  
(e) {u3 , u4 , u5 }  x
 

(k) S = y  ∈ R3 x = 6t, y = 4t some t ∈ R
 
 
(f) {u5 } 
z

 " # " # " #  
1 2 3 2 −6 5 3 −8 18
11. Let a1 = , a2 = , a3 = .
2 1 5 16. The matrix A =  −3 9 −1 −5 −1 −36 
 

0 0 4 8 −8 36
(a) Express a3 as linear combinations of a1 and a2 if possible.  
1 −3 0 0 1 4
has reduced form R =  0 0 1 0 −2 −1 .
 
(b) Is {a1 } a basis for R2 ? Why or why not? 0 0 0 1 0 5
(a) Choose a basis for Col(A) from the columns of A.
(c) Is {a1 , a2 , a3 } a basis for R2 ? Why or why not? (b) Choose a basis for Nul(A).
 
5 4 1 0 1 13
(d) Is {a2 , a3 } a basis for R2 ? Why or why not?
4 5 −1 0 1 14 
17. The matrix A = 
 

−4 −4 0 0 0 −12
3 2 1 0 1 7
12. Let u1 = (4, 2, 5), u2 = (3, −1, −2), and u3 = (6, 2, 0)  
1 0 1 0 0 1
(a) Is {u1 , u2 , u3 } a basis for R3 ? Justify. −1 0 0 2
0 1
reduces to R =  .
 
0 0 0 0 1 0
0 0 0 0 0 0
(b) Is it possible to express u3 as a linear combination of u1
(a) Choose a basis for Col(A) from the columns of A.
and u2 ? Justify without solving.
(b) Choose a basis for Nul(A).
 
6 −9 2 −12 1 8
(c) Is it possible to express the vector (9, 5, 2) as a linear com-  −6 9 5 54 2 −1 
bination of u1 , u2 , and u3 ? Justify without solving. 18. The matrix A = 
 

 8 −12 1 −26 0 9 
    0 0 3 18 3 3
2 1 0 −1  
1 −3/2 0 −4 0 1
13. Let A = 3 1 1 and v =  2 .
   
 0 0 1 6 0 1 
reduces to R = 
 
1 0 1 3  0

0 0 0 1 0 
(a) Is v in Nul(A)? Justify your answer. 0 0 0 0 0 0

Let a1 , a2 , a3 , a4 , a5 , a6 be the columns of A.

(b) Is v in Col(A)? Justify your answer.
(a) Choose a basis for Col(A) from the columns of A.
     
1 −1 0 2 0 (b) Express each column of A that is not in your basis as a lin-
14. Let A = 1 −1 0, v1 = 2 and v2 = 0. ear combination of your basis vectors.
     

0 0 0 0 2 (c) Find a basis for Nul(A).

 
(a) Find a basis for Col(A). 3 3 a c 1 e
19. The matrix  b 2 −8 6 f 15 
 

0 d 0 2 1 6
(b) Find a basis for Nul(A).  
1 0 4 −1 0 0
has reduced form  0 1 0 2 0 5 .
 
(c) Is v1 in Nul(A)? Justify your answer. 0 0 0 0 1 1
Find a, b, c, d, e, and f .
(d) Is v1 in Col(A)? Justify your answer.  
2 a 5 3 b c
20. The matrix  d 9 e −5 −1 −36 
 
(e) Is v2 in Nul(A)? Justify your answer.
0 0 4 f −8 36
 
1 −3 0 0 1 4
(f) Is v2 in Col(A)? Justify your answer. has reduced form  0

0 1 0 −2 −1 .


0 0 0 1 0 5
15. Let a1 = (2, 3, −1, 1), a2 = (−2, −3, 1, −1), Find a, b, c, d, e, and f .
a3 = (2, 3, 1, 5), a4 = (2, 3, 2, 7), a5 = (4, 6, 3, 12).
Find a basis for S = Span{a1 , a2 , a3 , a4 , a5 }. 21. Suppose A is n × m, Dim( Col(A)) = 2, Dim( Nul(A)) = 3
and Dim( Nul(AT )) = 4. Find n and m.

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22. Suppose A is an 5 × 8 matrix. What is the general (parametric) solution to Ax = b?
(a) What is the minimum nullity of A? 26. 
Suppose
  the
 general
 solution
  Ax = b is given by
 to
(b) Can the system Ax = 0 have a unique solution? x 1 1 0
y =2 + s  0  + t 1.
       
(c) What is minimum nullity of AT ?
z 3 −2 5
(d) Can the system AT x = 0 have a unique solution?
23. Suppose A is a 6 × 4 matrix, and that the nullity of AT is 3. (a) Find a non-zero solution to the homogeneous equation
(a) Find the nullity of A. Ax = 0

(b) Does the system Ax = 0 have a unique solution? (b) Find the general solution to Ax = 2b.
(c) Are the columns of A linearly independent? (Hint: A(2x)=2(Ax).)
       
24. Suppose A is 5 × 3 and the general −2 5 6 x
  solution
 to the equation
27. Let u =  1 , v = 0, w = −4, and x = y .
         
x 1 5 −2
Ax = b is given by y = 0 +s1+t 0 .
        0 1 3 z
z −3 0 1
Suppose A is 7 × 3, and that Nul(A) = Span{u, v}.
(a) Find the general solution to Ax = 0.
(a) Find the nullity of A.
(b) Find the rank of A.
    (b) Find the rank of A.
 −3
 5  
25. Suppose Nul(A) = Span  4  ,  0  for a matrix A,
   
  (c) Give the general solution of Ax = 0 in parametric form.
2 1
 
  (d) Give the general solution of Ax = Aw in parametric form.
6
and that u =  −1  is one particular solution to Ax = b.
 

2 A NSWERS ON NEXT PAGE .

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A NSWERS:

1. (a) v = −4u1 + 7u2 . (f) Yes. S = Span {(2, −3)}.

(b) k = 1/6.
2. (a) h = 5. (g) No. Not closed under SM or VA.
C-E: Let u = (0, 0, −1) and k = 2. The vector u ∈ S,
(b) b = 53 a1 − 13 a2 .
but ku = u + u ∈ / S.
3. h = 17.    
     1
 0 
 3 −4 

(h) Yes. S = Span 0 ,  1  .
     
4. (a) The plane is given by Span 1 .  0  .
     
2 −3
 
 
0 1
 
   (i) No. Not closed under SM or VA.
 −1/2 
  C-E: Let u = (1, 1, −1) and k = 2. The vector u ∈ S,
(b) The line is given by Span  7/6  .
 
  but ku = u + u = (2, 2, −2) ∈ / S.
1
 

5. 8x − 7y − 5z = 0 or any non-zero multiple.

   
 1
 0 

(j) Yes. S = Span  0  , 1 .
   
6. (a) LI. 7. (a) LI.  
−2 1
 
(b) LD. (b) LD.    
 6
 0 
(c) LD. (c) LD. (k) Yes. S = Span 4 , 0 .
   
 
0 1
 
(d) LI. (d) LI.
(e) LI. (e) LI. 11. (a) a3 = 73 a1 + 31 a2 .

(f) LD. (f) LI. 6 R2 . (Takes at least two vectors...)

(b) No, since Span{a1 } =
(c) No, since {a1 , a2 , a3 } is linearly dependent.
8. (a) LI. Plane. 9. (a) LI. Line. (...and no more than two.)
(b) LD. Plane. (b) LD. Line. (d) Yes, since {a2 , a3 } is linearly independent and spans R2 .
3
(c) LI. R . (c) LD. Plane 4 3 6
(d) LD. R .3
(d) LI. R .3 12. (a) Yes, since 2 −1 2 6= 0.
5 −2 0
(e) LD. Line. (e) LD. R3 .
(b) No, since {u1 , u2 , u3 } is linearly independent.
(f) LI. Plane.
(c) Yes, since Span{u1 , u2 , u3 } = R3 .
(g) LD. Plane
(h) LD. Point.
    13. (a) No. Av 6= 0.
 4
 −1 
(b) Yes. Ax = v is consistent.

10. (a) Yes. S = Span 1 ,  3  .
   
     
1 −1 0 2 0
 
6 0
 
14. Let A = 1 −1 0, v1 = 2 and v2 = 0.
     
(b) No. Not closed under VA or SM. C-E: Let u = (1, 0, 1).
0 0 0 0 2
u ∈ S, but 2u = u + u ∈
/ S.      
 1 
   1
 0 

(a) 1 (b) 1 , 0
     
(c) No. Not closed under VA. C-E: Let u = (1, 0, 0). Let    
0 0 1
   
v = (0, 1, 0). The vectors u and v are in S,
but u + v = (1, 1, 0) ∈
/ S. (c) Yes, Av1 = 0.
   
 −2
 3  (d) Yes, v1 = 2a1 , double the first column of A.
h i
(d) Yes. S = Nul 1 2 −3 = Span  1  ,  0  .
   
  (e) Yes, Av2 = 0.
0 1
 
(f) No, Ax = v2 has no solution.
    
(e) No. S is not closed under SM. C-E: Let u = (0, 1, 0) and  2 2 
 
k = −1. The vector u ∈ S,  3  3
 
15. Basis for S:   ,   .
   
but ku = −1(0, 1, 0) = (0, −1, 0) ∈
/ S. 
 −1 1 

 

1 5

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       
 2
 5 3   21. n = 6, m = 5.
16. (a) Basis for Col(A): −3 , −1 , −5
     
  22. (a) Max Rank of A = 5, so Min Nullity of A = 8 − 5 = 3.
0 4 8
 
      (b) No. Solution must have at least 3 parameters.
3 −1 −4 
(c) Min Nullity of AT is 0.

 
1  0   0 
 

 
     
      
(d) Yes. Solution is unique when Nullity of A is 0.
0  2   1 
 
(b) Basis for Nul(A):   ,   ,  
     


 0  0  −5  23. (a) 1.

      
 0  1   0  
(b) No. There will be one parameter in the solution.

 

 
0 0 1
      (c) No. There is a non-trivial solution to Ax = 0.
 5 4 1       

 4   5  1
 
 x 5 −2
17. (a) Basis for Col(A) :   ,   ,  
     
24. (a) y =s1+t 0 
     

 −4 −4 0 

 
 z 0 1
3 2 1
      (b) Rank of A = 1.
 −1 0 −1  

        x = 6 − 3s + 5t
    −2


 1 0 
y = −1 + 4s



 1  0  0 
     
 25.

(b) Basis for Nul(A) :  , ,  . 
z = 2 + 2s + t
 0  1  0 

     

       


  0   0   0 

 26. (a) (x, y, z) = (1, 0, −2), for example when s = 1 and t = 0.

 

0 0 1        
x 2 1 0
18. (a) Basis for Col(A) : {a1 , a3 , a5 }. (b) y =4 + s  0  + t 1.
       

(b) a2 = − 23 a1 , a4 = −4a1 + 6a3 , a6 = a1 + a3 . z 6 −2 5


 3/2
    
4 −1  27. (a) 2.

 

 1   0   0  (b) 1.

     

 

    
 0  −6 −1    
(c) Basis for Nul(A) :  , ,  .  x =
 −2s + 5t
  0   1   0 
     


      

 (c) y = s


  0   0   0  



z = t

 

0 0 1 
 x =
 6 − 2s + 5t
19. (a, b, c, d, e, f ) = (12, −2, 3, 1, 16, 5).
(d) y = −4 + s

20. (a, b, c, d, e, f ) = (−6, −8, 18, −3, −1, 8). 
z = 3+t

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