Indicate whether each statement is true or false. If it is false, provide a coun
ID: 3109733 • Letter: I
Question
Indicate whether each statement is true or false. If it is false, provide a counter-example or an explanation showing that it can be false. a. The det(2A) = 2det (A). b. If T:R'" rightarrow R" is a one-to-one linear transformation, then there are no distinct vectors u and v in R'" such that T(u -v) = 0. c. Let S_1 and S_2 be sets of vectors. If Span (S_1) = Span(S_2), then S_1 = S_2. d. If A is not square, then the rows of A are linearly dependent. e. If A and B are n times n matrices, then (A + B)(A - B) = A^2 - B^2. f. If A is n times n and det (A) = 5, then det(A^3) = 15. g. A plane in R^3 is a two-dimensional subspace.Explanation / Answer
a)
False
For an n/x matrix
det(2A)=2^ndet(A)
b.
True
Other wise u-v being a non zero vector would be in kernel of T but a one to one map has the kernel {0}
c.
False
eg. in R2
S1={(1,0),(0,1)} spans R2
S2={(1,0),(1,1)} also span R2 but S1 and S2 are two different sets
d.
False
Let, A be 3x2 matrix
Since it has 2 columns so its rank can be atmost 2 so it can have atmost 2 linearly independent rows hecnce the rows aare linearly dependent
e.
False
(A+B)(A-B)=A^2-AB+BA-B^2
If, AB=BA is not true which isn't in general then given statement is false
f.
False
det(A^3)=det(A)^3=3^3=27
g.
False
R3 has dimensino 2