In: Advanced Math

list as many of the parts of fundamental theorem of linear algebra
as you can. An nxn matrix A is invertible if and only if ....

The answer is in the pic. If any doubt still remained, let me know in the comment section. If this solution helped, please upvote to encourage us. Thanks :)

What are the two parts of the Fundamental Theorem of Linear Algebra? How can they be applied? Give 1 example problem for each part.

fundamtal theorem of linear algebra for dimensions of the
fundamental subspace of A

Abstract Algebra and Galois Theory
Show how to one can deduce the Fundamental Theorem of Galois
theory from the Artin's lemma.

According to the Fundamental Theorem of Algebra, every
nonconstant polynomial f (x) ∈
C[x] with complex coefficients has a complex root.
(a) Prove every nonconstant polynomial with complex coefficients
is a product of linear polynomials.
(b) Use the result of the previous exercise to prove every
nonconstant polynomial with real coefficients is a product of
linear and quadratic polynomials with real coefficients.

Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x] can be written into a product of linear polynomials in C[x].

(a) Are there matrices A,B∈Mn(R)A,B∈Mn(R) such that AB−BA=IAB−BA=I. (b) Suppose that A,B∈Mn(R)A,B∈Mn(R) such that (AB−BA)2=AB−BA(AB−BA)2=AB−BA. Show that AA and BB are commutable.

Let A ∈ Mn(R) such that I + A is invertible. Suppose that B = (I − A)(I + A)-1(a) Show that B = (I + A)−1(I − A)(b) Show that I + B is invertible and express A in terms of B.

Exercise 13. Let A = (aij)n ∈ Mn(R) where aij = cos(i + j) for i, j = 1, 2, . . . ,n. Find rank(A).

Exercise 11. Find the rank of matrix A where A, B and C

Exercise 2. In S8, write the following permutations into cyclic form, then determine their signature.(a) 85372164 (b) 87651234 (c) 12435687

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