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Q3 14 Points Consider the vector space P2(R). Let ri, r2, 13 be 3 distinct real...
Q3 14 Points Consider the vector space P2(R). Let T1, T2, T3 be 3 distinct real...
Q3 14 Points Consider the vector space P2(R). Let T1, T2, T3 be 3 distinct real numbers and 21, 22, az be three strictly positive real numbers. Define (p(x), q(x)) = Li_1 Qip(ri)q(ri) Q3.1 5 Points Show that this P2 (R) together with (-:-) is an inner product space. Please select file(s) Select file(s) Save Answer Q3.2 2 Points Give a counter example that (-, - ) is not an inner product when T1, 12, 13 are still distinct real...
Q5 8 Points Let A = -4 0 -3 2 3 0 -2 -1 3 Q5.1...
Q5 8 Points Let A = -4 0 -3 2 3 0 -2 -1 3 Q5.1 6 Points Compute PTA (x) (the characteristic polynomial of TA), Write monomials in descending order of their exponent, x^n for 3" and a/b for Example: 5/2 (polynomial of degree 0) -2x+1/3 (polynomial of degree 1) 2x^2-3x+12 (polynomial of degree 2) Enter your answer here give the eigenvalue(s) of TA (in ascending order if multiple exists, separated by a comma and with a blank after...
3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let...
3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let X be the set formed by these l points. Prove that there are l vector fields F1, F2,..., Fe, each defined on R2X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on RP X for all i = 1, ..., l. (ii) (“linearly independent”) If C1,C2, ..., Ce are real numbers such that the vector...
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c –...
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
Q3 14 Points Consider the vector space P2(R). Let T1, T2, T3 be 3 distinct real numbers and 21, 22, az be three strictly positive real numbers. Define (p(x), q(x)) = Li_1 Qip(ri)q(ri) Q3.1 5 Points Show that this P2 (R) together with (-:-) is an inner product space. Please select file(s) Select file(s) Save Answer Q3.2 2 Points Give a counter example that (-, - ) is not an inner product when T1, 12, 13 are still distinct real...
Q5 8 Points Let A = -4 0 -3 2 3 0 -2 -1 3 Q5.1 6 Points Compute PTA (x) (the characteristic polynomial of TA), Write monomials in descending order of their exponent, x^n for 3" and a/b for Example: 5/2 (polynomial of degree 0) -2x+1/3 (polynomial of degree 1) 2x^2-3x+12 (polynomial of degree 2) Enter your answer here give the eigenvalue(s) of TA (in ascending order if multiple exists, separated by a comma and with a blank after...
3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let X be the set formed by these l points. Prove that there are l vector fields F1, F2,..., Fe, each defined on R2X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on RP X for all i = 1, ..., l. (ii) (“linearly independent”) If C1,C2, ..., Ce are real numbers such that the vector...
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
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