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Linear Algebra
Do row operations preserve the linear independence among the columns of a matrix? How about the rows of a matrix?
Linear Algebra
Prove that 0v = 0, for all v in V: I put down: from property 5 and property 8 we know that: v = 1v = (1+0)v = 1v + 0v = v + 0v = v+0 =v. thus, -v+v = -v+(v+0v) = (-v+v) + 0v. He said it was right but not totally right, and to prove it for anything (general case), ,and something about expressing W in terms of v in order to do that. Anyone?
Linear Algebra
a) Show that the complex dot product vec(A).vec(B) = vec(B)^(conjugate transpose)[vec(A)] can be obtained by:
vec(A).vec(B) = trace(AB^(conjugate transpose)) = tr(AB^(conjugate transpose)).
We can therefore use the trace to define an inner product between matrices: <A,B>=
trace (AB^(conjugate transpose)).
b) Show that trace(AA^(conjugate transpose)) >= 0 for all A, so that we can use the trace to define a norm on matrices: ||A||^2_F = trace (AA^(conjugate transpose)). This norm is the Frobenius norm.
vec(A).vec(B) = trace(AB^(conjugate transpose)) = tr(AB^(conjugate transpose)).
We can therefore use the trace to define an inner product between matrices: <A,B>=
trace (AB^(conjugate transpose)).
b) Show that trace(AA^(conjugate transpose)) >= 0 for all A, so that we can use the trace to define a norm on matrices: ||A||^2_F = trace (AA^(conjugate transpose)). This norm is the Frobenius norm.
Linear Algebra
Suppose the V, W, and Z are vector spaces with bases a,b, and c, respectively. Suppose also that T is a linear transformation from V to W and U is a linear transformation from W to Z. Let A represent T with respect to the bases a and b, and let B represent U with respect to the bases b and c. Show that the matrix BA represents the linear transformation UT with respect to the bases a and c.
Linear Algebra
Show that the rank of a matrix C = AB is never greater than the smaller of the rank of A and the rank of B. Can it ever be strictly less than the smaller of these two numbers?
Linear Algebra
Let u and v be two non-zero N-dimensional complex column vectors. Show that the rank of the N by N matrix uv^(conjugate transpose) is one.
Linear Algebra
Show that rank(A+B) is never greater than the sum of rank(A) and rank(B).
Linear Algebra
Let A and B be M by N matrices, P an invertible M by M matrix, and Q an invertible N by N matrix, such that B = PAQ, that is, the matrices A and B are equivalent. Show that the rank of B is the same as the rank of A. (Show that A and AQ have the same rank).
Linear Algebra
Let A be an M by N matrix. When does A have a left inverse? When does it have a right inverse?
Linear Algebra
Prove that a square matrix is invertible if and only if it has a full rank.
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#339914 on Dec 2023
#339914 on Dec 2023