10- T(v) T(w -10 10 T(u) -10- Ay 10- T(v) T(w) T(u) 10 10 -10- 41
Use a property of determinants to show that A and AT have the same characteristic polynomial. NONE Start with det (AT-Al)-det (AT-AIT) = det(A-Al)T. Then use the formula det AT-det A. Start with det A1( -1 ) det AT. Then use the formula AAT I. Start with det (AAT). Use the formula det AB (det A)(det B) to write det (AAT) (det A)(det AT). Then use the formula AAT-1
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Let u and v be the vectors shown in the figure to the right, and suppose...
Problem 1. The figure below shows the vectors u, v, and w, along with the images...
Problem 1. The figure below shows the vectors u, v, and w, along with the images T(u) and T(v) to the right. Copy this figure, and draw onto it the image T(w) as accurately as possible. (Hint: First try writing w as a linear combination of u and v.) TV (u) Problem 2. Let u = | and v Suppose T : R2 + R2 is a linear transformation with 6 1 3) Tu = T(u) = -3 and T(v)...
Suppose T: ℝ3→ℝ2 is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are...
Suppose T: ℝ3→ℝ2 is a linear transformation. Let U and V be the
vectors given below, and suppose that T(U) and T(V) are as given.
Find T(3U+3V).
Suppose T: R->R2 is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(3U+3V). 5 5 6 T(V) 6 =n 2 -3 T(U) V = 3 -4 3 -4
Suppose T: R->R2 is a linear transformation. Let U and V...
101-2019-3-b (1).pdf-Adobe Acrobat Reader DC Eile Edit iew Window Help Home Tools 101-2019-3-b (1...
101-2019-3-b (1).pdf-Adobe Acrobat Reader DC Eile Edit iew Window Help Home Tools 101-2019-3-b (1) Sign In x Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y, x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V -> V such that U is not an...
Problem 13. For each of the following we are given two vectors u, we V and...
Problem 13. For each of the following we are given two vectors u, we V and a linear trans- formation from a vector space V to itself. Check if the given vectors are eigenvectors for the transformation. If yes, then find the corresponding eigenvalues. (a). V=P3, 7(p(x))=x?p"(x) — xp'(x), with u =2+3x? and w=x?. (b). V = Muj, T(A)=A+A”, with u=[?) and w=[; ?] (c). V = P2, L(P(x) p(x)dx + (x – 3)p'(x) with u = 100 and w=3+3x.
11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) =...
11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) = 12 - (a+d)X + ad-bc = 0. Show that p(A) = A - (a + d) A+ (ad - bc)1 = 0. 12. (adapted from 1.6 14) Suppose A has eigenvalues 0,0,3 with independent eigenvectors u, v,w. (a) Give the vectors span the nullspace and the column space. (b) Find a particular solution to Ax=w. Find all solutions. (c) Does w + u in...
41 and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be...
41
and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
Question 1 Question 2 Let u, v, w be three vectors in R4 with the property...
Question 1
Question 2
Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
Let u = (2,3), v = (-5, 6), and w = (9,0). (a) Draw these vectors...
Let u = (2,3), v = (-5, 6), and w = (9,0). (a) Draw these vectors in R2 y 10 10 10 5 -10 -5 5 10 -10 -5 10 -10 -5 10 -10 o y 10 -10 -5 10 -10 O X (b) Find scalars 1, and in such that w = 1,0 + 12v. (11.12) - -1,2
Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v,w. (a)...
Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v,w. (a) Give a basis for the nullspace and a basis for the column space. (b) Find a particular solution to Ax=y+w. Also, find all solutions to Ax=y+w.
Problem 1. The figure below shows the vectors u, v, and w, along with the images T(u) and T(v) to the right. Copy this figure, and draw onto it the image T(w) as accurately as possible. (Hint: First try writing w as a linear combination of u and v.) TV (u) Problem 2. Let u = | and v Suppose T : R2 + R2 is a linear transformation with 6 1 3) Tu = T(u) = -3 and T(v)...
Suppose T: ℝ3→ℝ2 is a linear transformation. Let U and V be the
vectors given below, and suppose that T(U) and T(V) are as given.
Find T(3U+3V).
Suppose T: R->R2 is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(3U+3V). 5 5 6 T(V) 6 =n 2 -3 T(U) V = 3 -4 3 -4
Suppose T: R->R2 is a linear transformation. Let U and V...
101-2019-3-b (1).pdf-Adobe Acrobat Reader DC Eile Edit iew Window Help Home Tools 101-2019-3-b (1) Sign In x Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y, x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V -> V such that U is not an...
Problem 13. For each of the following we are given two vectors u, we V and a linear trans- formation from a vector space V to itself. Check if the given vectors are eigenvectors for the transformation. If yes, then find the corresponding eigenvalues. (a). V=P3, 7(p(x))=x?p"(x) — xp'(x), with u =2+3x? and w=x?. (b). V = Muj, T(A)=A+A”, with u=[?) and w=[; ?] (c). V = P2, L(P(x) p(x)dx + (x – 3)p'(x) with u = 100 and w=3+3x.
11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) = 12 - (a+d)X + ad-bc = 0. Show that p(A) = A - (a + d) A+ (ad - bc)1 = 0. 12. (adapted from 1.6 14) Suppose A has eigenvalues 0,0,3 with independent eigenvectors u, v,w. (a) Give the vectors span the nullspace and the column space. (b) Find a particular solution to Ax=w. Find all solutions. (c) Does w + u in...
41
and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
Question 1
Question 2
Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
Let u = (2,3), v = (-5, 6), and w = (9,0). (a) Draw these vectors in R2 y 10 10 10 5 -10 -5 5 10 -10 -5 10 -10 -5 10 -10 o y 10 -10 -5 10 -10 O X (b) Find scalars 1, and in such that w = 1,0 + 12v. (11.12) - -1,2
Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v,w. (a) Give a basis for the nullspace and a basis for the column space. (b) Find a particular solution to Ax=y+w. Also, find all solutions to Ax=y+w.
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