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It is a Linear Algebra HW. Skip the questions that are noted “(NOT GRADED)” (questions 4, 5(a,b), and 10).

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Math 27: Linear Algebra

Homework 6

Due: Tuesday 10/20

Directions: Please solve the following problems by hand, showing all necessary work.For any proofs,

all steps should be clearly justified and use mathematically correct notation. Good luck!!!

h 1 i

h0i

h −2 i

h0i

1. Consider vectors ~a = −1 , ~b = 0 , ~c = 2 , and d~ = 0 .

1

0

0

0

~ ~

(a) Describe

h 1 i the span of S = {~a, b, ~c, d} in terms of the span of as few vectors in S as possible. (b)

~ Explain in each case.

Is ~e = 1 in the span of S? (c) Is ~c in the span of S2 = {~a, d}?

1

1 0 1

2. Consider matrix A =

. (a) Is the range of A all of R2 ? (b) Describe the nullspace

3 6 3

h1i

1 ] in the

of A as a span of one or more vectors. (c) Is ~y = 1 in the nullspace of A? (d) Is ~x = [ 15

1

range of A? If so, express ~x as a linear combination of the columns of A. Explain in each case.

0 1

3. Describe the nullspace and range for A = 0 2 using the span of as few vectors as possible.

0 3

4. Consider a 5 × 5 matrix A whose nullspace contains only ~x = ~0. (a) How many leading ones

(pivots) must the echelon form of A have in this case? (b) Does A~x = ~b (where ~b has 5 components)

always have a solution in this case? Explain. (NOT GRADED)

5. Mystery Matrices. (a and b only are NOT GRADED)

2

(a) Construct a 2 × 4 matrix whose nullspace only contains all linear combinations of

2 and

1

04

3 .

2

1

3

1

0

1

.

(b) Construct a 3 × 4 matrix whose nullspace only consists of all scalar multiples of

h1i

h0i

h1i

(c) Construct a 3 × 3 matrix whose column space has 1 and 3 and whose nullspace has 1 .

5

1

2

(d) Construct a 2 × 2 nonzero matrix whose nullspace is the same as its column space.

6. Determine a subset of the vectors in S that is a basis for the span of S, where S = {

h1i h2i h3i h1i

2 , 5 , 7 , 1 }.

1

7. (a) Explain why it is impossible to construct a matrix whose column space contains

h1i

h0i

h0i

1 , with nullspace containing 0 and 0 .

1

1

0

1

h1i

1

0

and

1

(b) If AB = 0, a matrix of all 0s, then the column space of B is contained in which fundamental

subspace corresponding to A? Give an example of A and B, where neither of these matrices have

all zero elements.

3

1 1 0

1 1 0

8. Consider the matrices A = 1 3 1 and U = 0 2 1 .

3 1 −1

0 0 0

Determine the dimension of: (a) the column space of A, (b) the column space of U , (c) the row

space of A, (d) the row space of U . (e) Which of these four subspaces of R3 are the same? Explain.

9. Determine if the following statements are true or false. Explain.

(a) If the columns of a matrix are linearly dependent, then so are the rows.

(b) The column space of a 2 × 2 matrix is the same as its row space.

(c) The column space of a 2 × 2 matrix has the same dimension as its row space.

(d) The set of all columns of a matrix are a basis for the column space of that matrix.

10. The relationship between the column space and linear systems. (i) If we append an

extra column ~b to the right of the rightmost column of matrix A, then the column space of the

resultant matrix is larger than the column space of A unless what condition holds? (ii) Give an

example where the column space gets larger and an example where it does not. (iii) Why is A~x = ~b

solvable exactly when the column space does not get larger? (NOT GRADED)

11. Who said that Subspaces are just science fiction? Consider the motion of a robot

created by Professor Barranca. Here each column of a matrix A corresponds to a routine which

moves the robot linearly in a specified direction by a specific distance together prescribed by the

corresponding column. The set of all columns composing A thus corresponds to all possible routines,

which may be linearly combined to yield additional paths of motion. In this context, provide a

physical interpretation of what it means for a vector to be in (a) the column space of A and (b)

the nullspace of A. (c) Suppose now that several of the routines are broken, with corresponding

columns removed from A, and the column space has consequently decreased in size. What does

this mean about the complexity (number of achievable paths) of the robot’s dynamics? If columns

were removed and instead the column space remained identical, so would the complexity of the

robot’s dynamics. For this reason, the number of linearly independent columns of a matrix (rank )

may measure its information content. The fact that many matrices containing user data, about

movie preferences for example, are of low rank is crucial for making accurate suggestions even with

very limited preference information.

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Tags:

linear combination

Consider vectors

Consider matrix

Scalar multiples

Construct a matrix

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