Department of Mathematics, University of Toronto
MAT223H1S - Linear Algebra I
Suggestions for Review: Final Exam
The following is more of a review sheet for the material after Midterm II than it is one for the ▯nal exam. I
would suggest using the review sheets for the two midterm exams along with this one, as review sheets for
the ▯nal exam.
I suggest before you go through this document that you read through all your class notes and/or the textbook
and, as you’re reading, solve questions as you encounter a new topic. You should at least attempt to solve
all the computational style questions from the textbook, then try some of the Find an Example questions
and some of the True or False questions. Once you’ve systematically worked through all the material from
the beginning, only then read this document. Organizing a carrying out your own review is a critical step
to understanding and mastering new material. It is much more valuable than going through someone else’s
review. You should be able to state theorems and de▯nitions precisely and make connections between various
ideas. You should be able to solve particular examples explaining you reasoning at every step. If you are
doing 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. Working
in small groups may worthwhile.
This may not cover every idea you need to master (not that I’m intentionally omitting something) but it is
intended to identify and solidify the main points and connections between various concepts.
I hope this helps. Good Luck!
Chapter 8: Orthogonality
Keywords: Dot product, norm (length) of a vector, orthogonal, orthogonal vectors, orthogonal complement,
orthogonal set, orthogonal basis, orthogonal projection onto a vector, orthogonal projection onto a subspace,
Gram-Schmidt process, orthonormal set, orthonormal basis, normalizing.
▯ De▯ne what it means for the set1fv ;::k;v g of vectors in R to be orthogonal. De▯ne what it means
for the set to be an orthogonal basis for a subspace W of R . What is the di▯erence between an
orthogonal basis and an orthonormal basis?
▯ Does every subspace of R have an orthonormal basis? What about the zero subspace? Given a basis
for a subspace W of R , how do you ▯nd an orthonormal basis? Describe the Gram-Schmidt procedure.
▯ What are the essential properties of a projection onto a subspace W of R . How do you calculate
projW (x) for any x 2 R ? What is pWo(x) if x 2 W? What is prWj(x) if x 2 W ?
2 3 2 3 2 3 2 3
n x1 x1 1 o ▯2
1(a) Consider the subspace W = 4x25 j x25 ▯ 0 5= 0 of R . Express the vector x =1 5 as the
x3 x3 ▯1 3
sum of a vector in W and a vector in W .
(b) Let W be a subspace of R and let x 2 R . Prove that x = pW(x) + proW ?(x).
1 of 4 2 3 2 3 2 3 2 3
( 2 1 1 1 )
6▯1 7 6 3 7 6 1 6 ▯2 7
2. Let ▯ = 4 5 4 5 4 5 4 5 .
▯1 3 0 3
▯1 ▯4 1 1
(a) Show that ▯ is an orthogonal basis for R .
(b) Normalize the vectors in ▯ to produce an orthonormal basis ▯ for R .
(c) Let x =6 2 7. Find [x] - the coordinate vector of x with respect to ▯ - and show that jjxjj = jj[x] jj.
4▯1 5 ▯ ▯
(d) Prove a more general version fo part (p)1 Let kv ;:::;v g be an orthonormal set of vectors in R and
let x = x v + ▯▯▯ + x v . Prove that jjxxj+ :::x .
1 1 k k 1 k
2 3 2 3 2 3
1 1 1
617 6▯2 7 6 1 7 4
3(a) Let 1 = 4 5, x2= 4 5, and 3 =4 5. Show that f1 ;2 ;3 g is an orthogonal subset of R and
1 1 1
1 0 ▯3
▯nd a fourth vector x such that fx ;x ;x ;x g is an orthogonal basis for R . To what extent is x
4 2 3 1 2 3 4 4
unique? (Hint: Let4x =6 7 and solve the system of equations de▯nei b4 x ▯x = 0 for 1 ▯ i ▯ 3.)
3(b) Prove a general version of 3(a): 1et2fx3;x ;x g be an orthogonal subset of R . Prove that you can
always ▯nd a fourth vect4r x such th1t 2x 3x 4x ;x g is an orthogonal basis for R . To what extent
is 4 unique? Prove your answer. (Hint: What is the rank of the coe▯cient matrix for the system of
linear equations de▯ned iy 4 ▯ x = 0 for 1 ▯ i ▯ 3?)
2 3 2 3 2 3 2 3
( 1 0 ) ( 1 3 )
6 7 6 7