A sample of 2,500 people was asked how many cups of coffee they drink in the morning. You are given the following sample informa
1 answer:
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Answer:
(B), (C) and (D) are true
Step-by-step explanation:
A) The mapping is the linear tranformation induced by the matrix A. The image is a vector space generated by the columns of the matrix A, and the range is the dimension of that subspace. So both concepts differ. This statement is False.
B) This statement is True by definition.
C) Col(A) is the set of vectors of R^m that are generated by the columns of A. The columns of A are obtained from multiplying A by the canonical base of . if
is any vector spanned by the columns of A, lets say
, where
are scalars and
is the
column of A, then Aw = v, with
. Reciprocally, any element of the form Ax is spanned by the columns of A. In fact,
, with
. This proves that (c) is True.
D) This is True. Let T be a linear transformation, and let ,
. Since T is a linear transformation, it maps 0 to 0, and we have:
- 0 ∈ ker(T) : T(0) = 0
- T(v+w) = T(v) + T(w) = 0 + 0 = 0
- T(λv) = λT(v) = λ*0 = 0
This proves that ker(T) is a vector space, because it is a vector subspace of .
E) An mxn matrix can only be multiplied (to its right) by a vector of , thus its null space lives in
, not
. This statement is False.
F) The equation Ax=b being consistent means that that particular element of b is on Col(A) . In order for Col(A) to be , the equation Ax = b needs to be consistent for every element
, not just one in particular. A matrix wiht its first column equal to b and the rest of its columns equal to 0, will satisfy that the equation Ax = b is consistent for that b, but Col(A) is <b>, and not
. This shows that (F) is False.