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Exam 1 Solved | Linear Algebra | MATH 203, Exams of Linear Algebra

George Mason University (GMU)Linear Algebra

Material Type: Exam; Class: Linear Algebra; Subject: Mathematics; University: George Mason University; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 02/10/2009

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Prof. Gabel Sign Name: _________________________Print Name__________________________________
Math 203 Matrix Algebra -- Exam I: Professor Gabel (March 28, 2002)
Directions: Do ALL of your work on these sheets of paper. No calculators are allowed. In order to maximize
the credit you receive, show all of your steps, write neatly and give some reasons. There are two types of
reasons: either by definition or by theorem. Make clear which you are using.
Remember, the honor code is to be observed on this exam. You are allowed one "crib" sheet on this exam.
[Suggestion: some of the problems say: show something. Be guided in your answer by how much space I have
left for your work. In particular, nearly all of these problems require little space to solve. Think a little, think a
bit more, and then start writing.] Problems marked NPC are No Partial Credit.
There are 150 points on this exam Make sure you have 2 sheets (double-sided) and put your name on each sheet.
Recall two definitions:
(a) A subspace of a vector space is a subset set which contains 0 and is closed under addition and scalar
multiplication.
(b) A collection of vectors is a basis for a vector space (or a subspace) if they are independent and span that
space.
(1) NPC Each of the questions below is worth 5 points (making 50 points total). Circle T for True, F for False
or circle blank if you wish to leave the problem blank.
SCORING: correct = +5, blank = +2 wrong = +0
T F blank Assume A is row equivalent to B. If Av=0 then it MUST be that Bv=0.
T F blank Assume A is row equivalent to B. If Av=0 then it MIGHT be that Bv=0.
T F blank Assume v1, v2, v3 are a basis for a vector space V. Then v1, v2 MUST also be a basis for V.
T F blank Assume v1, v2, v3 are a basis for a vector space V. Then v1, v2 MIGHT also be a basis for V.
T F blank If E and F are elementary matrices, then the product EF MUST also be an elementary matrix.
T F blank If E and F are elementary matrices, then the product EF MIGHT also be an elementary matrix.
T F blank Let A be a 3x2 matrix. Then A MUST have a right inverse.
T F blank Let A be a 3x2 matrix. Then A MIGHT have a right inverse.
T F blank If the columns of an mxn matrix span Rm then the rows MUST be independent.
T F blank If the columns of an mxn matrix span Rm then the rows MIGHT be independent.
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Download Exam 1 Solved | Linear Algebra | MATH 203 and more Exams Linear Algebra in PDF only on Docsity!

Math 203 Matrix Algebra -- Exam I: Professor Gabel (March 28, 2002)

Directions: Do ALL of your work on these sheets of paper. No calculators are allowed. In order to maximize

the credit you receive, show all of your steps, write neatly and give some reasons. There are two types of

reasons: either by definition or by theorem. Make clear which you are using.

Remember, the honor code is to be observed on this exam. You are allowed one "crib" sheet on this exam.

[Suggestion: some of the problems say: show something. Be guided in your answer by how much space I have

left for your work. In particular, nearly all of these problems require little space to solve. Think a little, think a

bit more, and then start writing.] Problems marked NPC are No Partial Credit.

There are 150 points on this exam Make sure you have 2 sheets (double-sided) and put your name on each sheet.

Recall two definitions:

(a) A subspace of a vector space is a subset set which contains 0 and is closed under addition and scalar

multiplication.

(b) A collection of vectors is a basis for a vector space (or a subspace) if they are independent and span that

space.

(1) NPC Each of the questions below is worth 5 points (making 50 points total). Circle T for True, F for False

or circle blank if you wish to leave the problem blank.

SCORING: correct = +5, blank = +2 wrong = +

T F blank Assume A is row equivalent to B. If Av=0 then it MUST be that Bv=0.

T F blank Assume A is row equivalent to B. If Av=0 then it MIGHT be that Bv=0.

T F blank Assume v 1 , v2, v 3 are a basis for a vector space V. Then v 1 , v 2 MUST also be a basis for V.

T F blank Assume v 1 , v2, v 3 are a basis for a vector space V. Then v 1 , v 2 MIGHT also be a basis for V.

T F blank If E and F are elementary matrices, then the product EF MUST also be an elementary matrix.

T F blank If E and F are elementary matrices, then the product EF MIGHT also be an elementary matrix.

T F blank Let A be a 3x2 matrix. Then A MUST have a right inverse.

T F blank Let A be a 3x2 matrix. Then A MIGHT have a right inverse.

T F blank If the columns of an mxn matrix span Rm^ then the rows MUST be independent.

T F blank If the columns of an mxn matrix span Rm^ then the rows MIGHT be independent.

(2) [5 pts] NPC Find the reduced echelon form of the matrix: 

____________

(3) [20 pts NPC] Consider the system of linear equations: x 1 - 3x 2 - 6x 4 - x 5 = 2

x 3 + 2x 4 - 3x 5 = -

Fill in the missing vectors so that the expression below gives all the solutions to the above system

x 2 x 4 x 5.

(4) [10 pts NPC] Let A =

. Find its inverse. [I suggest you check your answer.]

If you row reduce (I A), the resulting 3x3 matrix on the right is the inverse:

(5) [10 pts] Consider the vector space P 4 , the set of all polynomials of degree  4 in the variable t.

(a) Show t  Span {1 + t + t^2 , -1 + t - t^2 }.

Since t = (1/2)[ 1 + t + t^2 ] + (1/2) [-1 + t - t^2 ], t is indeed a linear combination of 1 + t + t^2 and -1 + t - t^2 ,.and so

t is in Span {1 + t + t^2 , -1 + t - t^2 }

(b) Show t, 1 + t + t^2 , -1 + t - t^2 are dependent.

We need to show some non-trivial linear combination of these three vectors is 0. Well, the answer to part (a)

shows that (-1) t + (1/2)[ 1 + t + t^2 ] + (1/2) [-1 + t - t^2 ] = 0. so we are done.

(8) [15 pts] Consider the linear transformation T: R^3  R^3 given by T

x y x y z y x

(a) [NPC] Find the matrix of this transformation.

By definition , this matrix is: [T(e 1 ) T(e 2 ) T(e 3 )], which equals:

(b) Is T one-to-one? Explain, making it clear from your answer that you know what one-to-one means.

T is not one-to-one because there is NOT a pivot in teach column of the matrix of T. OR, it is not one-to-

one as the both of the following vectors are clearly sent to 0 under T:

and

(c) Is T onto? Explain, making it clear from your answer that you know what onto means.

No, T is not onto, for the matrix of T does not have a pivot in each row. OR, T is not onto because, for

example,

is not in the image of T as its third component is not zero while, clearly all vectors in the image of T

have a 0 in the third component.

(9) [10 pts] Let A =

and consider the function T: R^3  R^3 given by T(v)=Av.

(a) Find one vector whose span equals nul A. Explain, briefly, why your answer is correct.

[Recall nul A equals {x  R^3 such that A(x) = 0.} ]

We have a procedure for finding a basis for nul A: it's called row reduction and then using the ideas of

problem (3) of this exam. If you do that, x 3 will be the free variable and you'll get that the solution set is

x 3

. Thus,

.is a basis for nul A.

(b) Find two vectors whose span equals col A. Explain, briefly, why your answer is correct.

[Recall col A = Span {columns of A} ]

Well, col A is the span of all the columns of A, but you don't need the third: you need just the first two. They

are clearly (out of space!) independent. OR, use the theorem that the pivot columns of A are a basis for the

column space of A. The pivots are clearly in columns 1 and 2, so the first two columns are a basis for col A.