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Suppose that T: R²→R³ is a linear transformation such that and Determine for any in R².

其他申論題

109中興電機 (15 pts) The eigenvalues of A and AT are the same, because det(A- λ l) -det(A- λ l)T= det(AT- λ l). By coming up with a 2 X 2 counter-example, show an example that the cigenvectors of A and ATneed not be the same.

106中興電機 (10pts) Suppose A is a real and symmetric matrix with order n, and has a repeated eigenvalue. Then for every i in {1,2, ... n}, there exists an eigenvector whose ith component is 0.

105中興電機 (10pts) Suppose A and B are two matrices with size m✕n and n✕m resbectively. Show that the nonzero eigenvalues of AB and BA are the same.

Let .Show that is linear combination of A and I2.

Let T be a linear operator on R3 such that T, Please find the standard matrix of T.

Let Y be the vector space of 2x2 mattices with real entries, and P3 the vector space of real polynomials of dogree 3 or less. Detine the linear transformation T: V → P3 by == 2a+(6-d)t-(a+c)x2+(a+6-c-d)x3. Find the rank and nullity of T.

(a). Prove that if ker T= {0}; then T is one-to-one.(7 pts)

(b). Suppose T is one-to-one and {u1,... ,uk} is a linearly independent set of vectors in U. Prove that [T(u1),....,T(uk)] is a linearly independent set of vectors in V.(7 pts)

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