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Unformatted text preview: n TA : x → Ax is Rm .
True or False? khan (sak2454) – HW05 – gilbert – (57245)
1. FALSE correct 005 2. TRUE
Explanation:
When A is m × n, the matrix product Ax
is only deﬁned for x in Rn , and the product
is then a vector in Rm . Thus TA : x → Ax
maps Rn to Rm . So the Range of TA is the set
{ TA (x) : x in Rn }
of vectors in Rm . But this need not be all of
Rm .
For example, since
1
0 1
0 x1
x2 = x1 + x2
,
0 the range of TA does not contain any vector
in R2 whose second entry is nonzero.
Consequently, the statement is
FALSE . 2
10.0 points If A is an m × n matrix, then the range of
the transformation
T : Rn → Rm , TA : x → A x , is the set of all linear combinations of the
columns of A.
True or False?
1. TRUE correct
2. FALSE
Explanation:
By deﬁnition, the range of TA : x → A x
is the set
{Ax : x in Rn } .
But when A = [ a1 a2 . . . an ] , x1 x2 x = . ,
.
. xn 004 10.0 points Every linear transformation T : Rn → Rm
is a matrix transformation.
True or False?
1. TRUE correct
2. FALSE
Explanation:
If T : Rn → Rm is a linear transformation,
and A is t...
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 Fall '14

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