Linear Algebra

Section III. Basis and Dimension 117
to get that c1 =3andc2 =1/2. Then we have this.
� �
3
RepB(�v) =
−1/2
Here, although we’ve ommited the subscript B from the column, the fact that
the right side it is a representation is clear from the context.
The up side of the notation and the term ‘coordinates’ is that they generalize
the use that we are familiar with: in Rn and with respect to the standard
basis En, the vector starting at the origin and ending at (v1,...,vn) has this
representation.
⎛ ⎞ ⎛ ⎞
v1
vn
v1
⎜
RepEn ( ⎝ .
⎟ ⎜
. ⎠) = ⎝ .
⎟
. ⎠
Our main use of representations will come in the third chapter. The definition
appears here because the fact that every vector is a linear combination of
basis vectors in a unique way is a crucial property of bases, and also to help make
two points. First, we put the elements of a basis in a fixed order so that coordinates
can stated in that order. Second, for calculation of coordinates, among
other things, we shall want our bases to have only finitely many elements. We
will see that in the next subsection.
Exercises
� 1.16 Decide if each is a basis for R 3 � � � � � � . � �
1 3 0
1
� �
3
� �
0
� �
1
� �
2
(a) 〈 2 , 2 , 0 〉 (b) 〈 2 , 2 〉 (c) 〈 2 , 1 , 5 〉
3
� �
0
1
� �
1
1
� �
1
3 1
−1 1 0
(d) 〈 2 , 1 , 3 〉
−1 1 0
� 1.17 Represent � � the� vector � � with � respect to the basis.
1 1 −1
(a) , B = 〈 , 〉⊆R
2 1 1
2
(b) x 2 + x 3 , D = 〈1, 1+x, 1+x + x 2 , 1+x + x 2 + x 3 〉⊆P3
⎛ ⎞
0
⎜−1⎟
(c) ⎝
0
⎠, E4 ⊆ R
1
4
1.18 Find a basis for P2, the space of all quadratic polynomials. Must any such
basis contain a polynomial of each degree: degree zero, degree one, and degree
two?
1.19 Find a basis for the solution set of this system.
x1 − 4x2 +3x3 − x4 =0
2x1 − 8x2 +6x3 − 2x4 =0
� 1.20 Find a basis for M2×2, the space of 2×2 matrices.
vn
En