Eureka Math Algebra I Study Guide

120 | eureka Math algebra I Study guIde

● (^) Matrix An mn ́ matrix is an ordered list of nm real numbers, a 11 , a 12 ,.. .,
a 1 n, a 21 , a 22 ,.. ., a 2 n,.. ., am 1 , am 2 ,.. ., amn, organized in a rectangular array of m rows and n
columns:
aa a
aa a
aa a
n
n
mm mn
11 12 1
21 22 2
12









é

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. The number aij is called the entry in row i and column j.

● (^) Matrix Difference Let A be an mn ́ matrix whose entry in row i and column j is aij,
and let B be an mn ́ matrix whose entry in row i and column j is bij. Then, the matrix
difference AB- is the mn ́ matrix whose entry in row i and column j is abij- ij.
● (^) Matrix Product Let A be an mn ́ matrix whose entry in row i and column j is aij, and let
B be an np ́ matrix whose entry in row i and column j is bij. Then, the matrix product AB
is the mp ́ matrix whose entry in row i and column j is abij 11 ++abij 22 +abin nj.
● (^) Matrix Scalar Multiplication Let k be a real number, and let A be an mn ́ matrix
whose entry in row i and column j is aij. Then, the scalar product kA× is the mn ́
matrix whose entry in row i and column j is ka× ij.
● (^) Matrix Sum Let A be an mn ́ matrix whose entry in row i and column j is aij, and let B be
an mn ́ matrix whose entry in row i and column j is bij. Then, the matrix sum AB+ is the
mn ́ matrix whose entry in row i and column j is abij+ ij.
● (^) Modulus The modulus of a complex number z, denoted z, is the distance from the origin
to the point corresponding to z in the complex plane. If za=+bi, then za=+^22 b.
● (^) Network Diagram A network diagram is a graphical representation of a directed graph
where the n vertices are drawn as circles with each circle labeled by a number 1 through
n and the directed edges are drawn as segments or arcs with the arrow pointing from the
tail vertex to the head vertex.
● (^) Opposite Vector For a vector



v represented by the directed line segment AB



, the opposite

vector, denoted -



v, is the vector represented by the directed line segment BA



. If





v

v

v

vn

=

é

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ê

ê

ê

ê

ù

û

ú

ú

ú

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1

(^2)
in Rn, then -=

-

-

-

é

ë

ê

ê

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ê

ù

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ú

ú

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ú





v

v

v

vn

1

(^2).
● (^) Polar Form of a Complex Number The polar form of a complex number z is
ri(cos()qq+ sin)() where rz= and q=arg(z).
● (^) Position Vector For a point P(v 1 , v 2 ,.. ., vn) in Rn, the position vector



v, denoted by

v

v

vn

1

2



é

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ê

ê

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ê

ù

û

ú

ú

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or

⟨v 1 , v 2 ,.. ., vn⟩, is a free vector



v that is represented by the directed line segment OP



from

the origin O(0, 0, 0,.. ., 0) to the point P. The real number vi is called the ith component of

the vector



v.