Math 212: Linear Algebra Name:
The goal of this worksheet is to familiarize your with some basic ideas from linear algebra. By the
end of the worksheet you should feel comfortable working with the following notions:
•vectors, the length of vectors and how to mutiply them
•a matrix, its transpose, conjugate and adjoint
•adding and multiplying matrices
•the zero matrix and the identity matrix
(1) Matrices
An m×nmatrix is a rectangular array of numbers (real or complex) with mrows and n
columns.
A=
a11 a12 · · · a1n
a21 a22 · · · a2n
.
.
..
.
..
.
.
am1am2· · · amn
This is sometimes written as A= (aij) for short. We will be concerned mostly about
square, that is n×n, matrices and vectors, that is n×1 matrices. Two matrices are equal
if all of the entries are equal.
(a) Write down your favorite 3 ×2 matrix that contains at least two complex entries and
call it B.
(b) The transpose of a matrix A= (aij )=(aji) = AT. All of the entries get reflected
across the diagonal line from the top left corner to the bottom right corner. So what
was the a24 entry now becomes the a42 entry. Write down BT, where Bis the matrix
from the previous part.
(c) We can also find the complex conjugate of a matrix by conjugating each entry. ¯
A=
(¯aij). Write down ¯
B.
(d) Conjugating and transposing a matrix gives the adjoint, A∗=¯
AT. Calculate B∗.
(2) Arithmetic with matrices We can add, subtract and multiply matrices as long as the
sizes of the matrices match up in appropriate ways.
(a) If αis a number (real or complex) then we can multiply a matrix A(aij) by αby
multiplying each entry by α, that is αA= (αaij ). Pick a complex number, call it β,
and calculate βB.
(b) If we have two matrices of the same size then we can add them entry by entry to get
a new matrix. So for A= (aij ) and C= (cij ), the matrix A+C= (aij +cij ). Write
down your next favorite matrix of the appropriate size, call it D, and add it to B.
(c) To subtract Cfrom Awe take (−1)Cand then add it to A. So A−C=A+ (−1)C.
Calculate B−D.
(d) The zero matrix is the matrix with all entries equal to 0 and is denoted 0. What is
B+0?
(e) Multiplying matrices is a bit more complicated. If we want to multiply Aby Cto
get the product AC then Amust have then same number columns as Chas rows. In
practice it is clear if it is possible to multiply to matrices or not. The product AC =E
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