We denote by Mn times m the collection of all m times n matrices. Show that M n

Question

We denote by Mn times m the collection of all m times n matrices. Show that M n times m is a vector space. Let A be an n times n matrix. We define the trace of A, trace (A), to be the sum of the diagonal elements of A. For example, if A = [2 3 2 1 -1 3 3 3 3] then trace (A) = 2 + (- 1) + 3 = 4. Show that the collection of all 2 times 2 matrices A such that trace (A) = 0 is a subspace of M2 times 2. Recall that Pn is the vector space of all polynomials of degree less than or equal to n, and that the dimension of Pn is n + 1. Let Q be the collection of all polynomials q(t) in P3 such that q(7) = 0. That is Q is the collection of polynomials of degree less than or equal to 3, which have 7 as a root. Show that Q is a subspace of P3. Note that p(t) = t - 7, q(t) = t2 - 49 and r(t) = t3 - 343 are in Q. Show that B = {p,q,r} is a basis for the subspace Q. What is the dimension of Q? Let R be the collection of all polynomials q(t) in P3 such that q(7) = 3. Is R a subspace of P? Why or why not? Consider the vector x = [3 2 -1] in R3. Let H be the set of vectors in R3 which are orthogonal to x. That is, H is all vectors y such that x . y = 0 (recall that the dot product was defined in Assignment 6). Show that H is a subspace of R3. What is the dimension of H? (Hint: First explain why dim H = 3. Now, how large can a collection of linearly independent vectors in H be? Try and find a largest family of linearly independent vectors.)

Explanation / Answer

1)multiplication M(n*n) remains the same we know when two (n*n) matrices are added or multiplied we get a (n*n) matrice only so it is a subspace

2)|3 2|

|-1 3| 2*2 matrix this is trace a= 0

3) Pn = n+1

q is subspace of P3= 4

q(t) = t^2 - 49

q(7)= 0

let's t = 7

p(t)= t-7 , and r(t)= t^3-343

p(7) and r(7) also 0

4)

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