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  1. B

    Matrix and basis

    Consider a) f1=1, f2=sinx , f3=cosx b) f1=1, f2=ex , f3=e2x c)f1=e2x , f2=xe2x f3=x2e2x in each part B={f1,f2,f3} is a basis for a subspace V of the vector space. Find the matrix with respect to B of the differentiation operator D:V→V
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    Inner product of polynomials

    [-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.
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    Skew symmetric matrix

    what about thinking of rank-nullity theory since the dimension of this skew-symmetric matrix=n(n-1)/2 but how to calculate the dim of the AX=0
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    Skew symmetric matrix

    how can we prove that the rank of skew symmetric matrix is even i could prove it by induction is there another way
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    Isomorphic linear space

    two linear spaces S and S1 over F are isomorphic if and only if there is a one-to-one correspondence x↔ x1 between the elements x \in S and x1 \in S1 such that if x ↔ x1 and y ↔ y1 then x+y ↔ x1+y1 and ax ↔ ax1 (y \in S , y1 \in S1, a \in F). prove that two finite -dimensional spaces are...
  6. B

    Denote standard inner product

    if A \in C nxn,show that (x,Ay)=0 for all x,y \in C[n], then A=0 (x,Ay) denote standard inner product on C[n]
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    Rank of matrix

    check that, for any nxn matrices A,B then rank(AB) (> or =) rank A +rank(B)-n
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    Show that if all the row sums of a matrix A belong to C (nxm) are

    show that if all the row sums of a matrix A belong to C (nxm) are zeroes, then A is singular. Hint. Observe that Ax=0 for x=[1 1 ....1]T