Let T: R3 → R2 be a linear transformation defined by T(x,y,z) = (3x + 2y – 4z, x - 5y + 3z). Find the matrix of T relative to the basis (1,1,1),(1,1,0),(1,0,0) of R3 and (1,3), (2,5) of R2
Answers
Answer:
y = c
1
v
1
+ c
2
v
2
+ ... + c
n
v
n
for some unique bi
’s and c
i
’s in . Then, by definition of T, we have
T(x) = b1
w1
+ b2
w2
+ ... + bn
wn
T(y) = c
1
w1
+ c
2
w2
+ ... + c
n
wn
T(x) + T(y) = (b1
w1
+ b2
w2
+ ... + bn
wn
) + (c
1
w1
+ c
2
w2
+ ... + c
n
wn
)
= (b1
+ c
1
)w1
+ (b2
+ c
2
)w2
+ ... + (bn
+ c
n
)wn
However, x + y = (b1
v
1
+ b2
v
2
+ ... + bn
v
n
) + (c
1
v
1
+ c
2
v
2
+ ... + c
n
v
n
)
= (b1
+ c
1
)v
1
+ (b2
+ c
2
)v
2
+ ... + (bn
+ c
n
)v
n
T(x + y) = (b1
+ c
1
)w1
+ (b2
+ c
2
)w2
+ ... + (bn
+ c
n
)wn
,
again by definition of T. Hence, T(x + y) = T(x) + T(y). Next, for any scalar c ,
c x = c(b1
v
1
+ b2
v
2
+ ... + bn
v
n
) = (cb1
)v
1
+ (cb2
) v
2
+ ... + (cbn
)v
n
T(cx) = (cb1
)w1
+ (cb2
)w2
+ ... + (cbn
)wn
= c(b1
w1
) + c(b2
w2
) + ... +c(bn
wn
)
= c(b1
w1
+ b2
w2
+ ... + bn
wn
)
= cT(x)
Hence T is a linear transformation.
To prove the uniqueness, let L : V W be another linear transformation satisfying
L(v
1
) = w1
, L(v
2
) = w2
, ..., L(v
n
) = wn
If v V, then v = a1
v
1
+ a2
v
2
+ ... + an
v
n
, for unique scalars a1
, a2
, ..., an . But then
L(v) = L(a1
v
1
+ a2
v
2
+ ... + an
v
n
)
= a1 L(v
1
) + a2 L(v
2
) + ... + an
L(v
n
) ( L is a L.T.)
= a1
w1
+ a2
w2
+ ... + an
wn
= T(v)
L = T and hence T is uniquely determined.
EXAMPLE 14 Suppose L :
3
2
is a linear transformation with
L([1, –1, 0]) = [2, 1], L([0, 1, –1]) = [–1, 3] and L([0, 1, 0]) = [0, 1].
Find L([–1, 1, 2]). Also, give a formula for L([x, y, z]), for any [x, y, z]
3
.
[Delhi Univ. GE-2, 2017]
SOLUTION To find L([–1, 1, 2]), we need to express the vector v = [–1, 1, 2] as a linear
combination of vectors v
1
= [1, –1, 0], v
2
= [0, 1, –1] and v
3
= [0, 1, 0]. That is, we need to find
constants a1
, a2
and a3
such that
v = a1 v
1
+ a2 v
2
+ a3 v
3
,
which leads to the linear system whose augmented matr