Department of Mathematics, University of TorontoMAT223H1S - Linear Algebra I

Suggestions for Review: Midterm Exam I

I suggest before you go through this document that you read through all your class notes and/or the textbookand, as you’re reading, solve questions as you encounter a new topic. You should at least attempt to solveall the computational style questions from the textbook, then try some of the Find an Example questionsand some of the True or False questions. Once you’ve systematically worked through all the material fromthe beginning, only then read this document. Organizing a carrying out your own review is a critical stepto understanding and mastering new material. It is much more valuable than going through someone else’sreview. You should be able to state theorems and definitions precisely and make connections between variousideas. You should be able to solve particular examples explaining you reasoning at every step. If you aredoing a proof and you are using a theorem to help complete the proof, you should state the theorem precisely.Writing something like ”By Theorem in the textbook, yadda, yadda, yadda...” is not acceptable. Workingin small groups may worthwhile.

This may not cover every idea you need to master (not that I’m intentionally omitting something) but it isintended to identify and solidify the main points and connections between various concepts.

I hope this helps. Good Luck!

Chapter 1: Systems of Linear Equations

Keywords: Linear equation, systems of linear equations, homogeneous system of linear equations, solution,set of solutions, trivial solution, non-trivial solution, consistent, inconsistent, elementary row operations,equivalent systems, augmented matrix, Gaussian elimination, leading variable, free variable, parameter,echelon form, reduced echelon form.

• Explain why Gaussian elimination works as a method for solving systems of linear equations.

• When you row-reduce an augmented matrix [A | b ] to solve a system of linear equations, why can youstop when the matrix is in echelon form? How does the echelon form help you decide if the system isinconsistent or consistent? If it is consistent, how does it help you decide if you have a unique solutionor infinitely many solutions.

• Why are homogeneous systems of linear equations always consistent? How can you use the echelonform to determine whether there are non-trivial solutions and, if there are, how many parametersthere are in the general solution? Is there any case where you know (without row-reducing) that ahomogenous system is guaranteed to have a non-trival solution?

• How is reduced echelon form different from echelon form?

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Practice Problems

1. Consider the system of linear equations

x1 + 2x2 + x3 + 2x4 + x5 = 22x1 + x2 + 3x3 + 5x4 + 5x5 = 7

3x1 + 6x2 + 4x3 + 9x4 + 10x5 = 11x1 + 2x2 + 4x3 + 3x4 + 6x5 = 9.

Find all solutions to the system and express your answer in parametric form.

2. Given the system of linear equations

x1 + x2 + 3x3 = c,

cx1 + x2 + 5x3 = 4,

x1 + cx2 + 4x3 = c.

(a) For what values of c does the system have (i) no solutions, (ii)infinitely many solutions, (iii) a uniquesolution.

(b) Find the set of solutions for the value of c for which the system has infinitely many solutions.

Chapter 2: Euclidean Space

Keywords: Vector addition, scalar multiplication, linear combination, span, spanning set, vector equation,matrix equation, linear independence, linear dependence.

• State the formal definition of linear independence and linear dependence. Why is linear independenceimportant when looking at spanning sets?

• Explain how you use information about consistency of systems of equations and uniqueness of solutionsin determining is a vector belongs to the span of a given set of vectors and in testing if a set is linearlyindependent.

• Explain how you use information about consistency of systems of equations in testing if Rn is equal tothe span of some set of vectors in Rn.

Practice Problems

3. Let W = Span

{1302

,

1101

,

1011

}

. For what value of c is

010c

∈W?

4. Let A be a 3× 3 matrix with linearly independent columns.

(a) Explain why the reduced echelon form of A is

I =

1 0 00 1 00 0 1

.

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(b) Let A be a 3 × 5 matrix such that the first three columns are linearly independent. Explain why theleading 1’s in the reduced echelon form of A must be in the first three columns. (Hint: If this were nottrue, what would that tell you about part (a)).

5. Suppose that {v1, v2, . . . , vk} is a linearly independent set in Rn. Prove or disprove: The set {v2, v3, . . . , vk}cannot span Rn.

6. Suppose A is n × n such that the systems of linear equations represented by Ax = b has an infinitenumber of solutions. Prove or disprove: The columns of A are linearly dependent.

Chapter 3: Matrices

Keywords: Linear transformations, domain, co-domain, range, image of a vector x under T , 1-1, onto,matrix equality, matrix addition, scalar multiplication, transpose, matrix multiplication, additive identity,identity matrix, zero matrix, symmetric matrix, inverse, invertible matrix, non-singular, singular.

• Define what it means for a function T : Rm → Rn to be a linear transformation and explain how youcan always find a unique n × m matrix A such that T (x) = Ax. Explain how you use informationabout consistency of systems of equations and uniqueness of solutions in determining if T is onto andin determining if T is 1-1. What do the independence of the rows and/or columns of A tell you aboutT? Repeat the above in the special case when n = m.

• How do you determine the product of two matrices A and B? What condition(s) must be satisfied bythe sizes of A and B for the product to be defined?

• Explain the procedure for determining the inverse of a matrix. Explain why it might not produce andinverse for some matrix A (and that’s ok).

Practice Problems

7. Suppose that A is a 3× 3 matrix, AT + 5I is nonsingular, and

(AT + 5I)−1 =

1 3 00 −1 02 2 1

.

Find A. Here I is the matrix in question 4(a).

8. Let T : R2 → R2 be a linear transformation such that T( [1

2

] )=[73

], and

T( [3

4

] )=[−11

]. Find T

( [xy

] )for all

[xy

]∈ R2.

9. Let

A =

1 −5 4 −2 21 −6 5 −3 2−2 11 −8 5 −2

and let T : R5 → R3 be the linear transformation defined by T (x) = Ax for all x ∈ R5. Show that Tis onto but not 1-1.

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10. Let T : R3 → R3 be the linear transformation defined by

T(x1

x2

x3

) =

x1 + 2x2 + x3

x1 + 2x2 + 2x3

3x1 + 3x2 + x3

.

Show that T is invertible and find T−1(x1

x2

x3

).

11. Find all values of c such that the matrix A =

2 5 2 40 1 −1 10 1 4 21 0 1 c

is invertible.

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