Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar...
Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar multiplication is a finite
dimensional vector space
with dim GLm,n(R) = mn.
Show that if V and W be finite dimensional vector spaces with
dim V = m and dim W = n, B a basis for V and C a basis for W
then
hom(V,W)-----MatB--->C(-)-------->
GLm,n(R) is a bijective linear transformation. Hence or
otherwise, obtain dim hom(V,W).
Thank you!
Let A be a matrix with m distinct, non-zero, eigenvalues. Prove
that the eigenvectors of A...
Let A be a matrix with m distinct, non-zero, eigenvalues. Prove
that the eigenvectors of A are linearly independent and span R^m.
Note that this means (in this case) that the eigenvectors are
distinct and form a base of the space.
Given A is a mxn matrix with dim(N(A)) if u=(α(1), α(2),...,
α(n))^T ∈N(A). Prove that α(1)a(1)+α(2)a(2)+...+α(n)a(n)=0,...
Given A is a mxn matrix with dim(N(A)) if u=(α(1), α(2),...,
α(n))^T ∈N(A). Prove that α(1)a(1)+α(2)a(2)+...+α(n)a(n)=0, where
a(1), a(2),..., a(n) are columns of A.
Now suppose that B is the matrix obtained from A by performing
row operations: Show that α(1)b(1)+α(2)b(2)+...+α(n)b(n)=0, where
b(1), b(2),..., b(n) are columns of B.
Show that the converse is also true.
1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If...
1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...