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- For each linear transformation f :V W, find the associated...
know how to find the matrix representation [T]5 for a linear transforma- tion T V W...
know how to find the matrix representation [T]5 for a linear transforma- tion T V W with respect to bases a, B for V, W, respectively. know how to use the matrix representation [T5 and the coordinate map- pings R of T W to find bases for the kernel and image V, :Rm -> given two bases a, from a coordinates to 3 coordinates for Rn, know how to find the change of basis matrix
know how to find the...
Determine whether or not the following transformation T :V + W is a linear transformation. If...
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Find the matrix [T], p of the linear transformation T: V - W with respect to...
Find the matrix [T], p of the linear transformation T: V - W with respect to the bases B and C of V and W, respectively. T:P, → P, defined by T(a + bx) = b - ax, B = {1 + x, 1 – x}, C = {1, x}, v = p(x) = 4 + 2x [T] C+B = Verify the theorem below for the vector v by computing T(v) directly and using the theorem. Let V and W...
6. Let T P2 P be a linear transformation such that T P2P2 is still a...
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
4. Consider the trace on M3x3 restricted to the subspace Z3x3: (Note: this is still a linear tran...
4. Consider the trace on M3x3 restricted to the subspace Z3x3: (Note: this is still a linear transformation, when restricted to Z3x3.) Find the matrix [trlss of this linear transformation with respect to the bases B and S. Here S is the standard basis for R. (a) What is the size of [tr]ss? (b) Compute [trss (c) Check that [tr]SBL]B--[tr(L)]s.
4. Consider the trace on M3x3 restricted to the subspace Z3x3: (Note: this is still a linear transformation, when restricted...
Consider the linear transformation T from V = P2 to W = P2 given by Tao...
Consider the linear transformation T from V = P2 to W = P2 given by Tao + ayt + azt) = (-63, + 2a, + 3a2) + (2a, + 4aq + 2az)t + (220 + 2a, + 3a2)2 Let F = (F1, F2, f3) be the ordered basis in P2 given by f(t) = 1, 72(t) = 1 + t, f3(t) = 1 +t+2 Find the coordinate matrix (TFF of T relative to the ordered basis Fused in both V...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
For each transformation below, find the closed form of the transformation. 1) Let T be a...
For each transformation below, find the closed form of the transformation. 1) Let T be a linear transformation from R$ to M22 (R) [i Let B=1 0:00 [. :] [11] [12] [0 ] Let C= 12 41 -17 -5 65 -27 92 Let M = be the matrix transformation of T from basis B to C 17 58 -15 -51 81 The closed form of the transformation is Tb 3-1 2) Let T be a linear transformation from P3(R) to...
know how to find the matrix representation [T]5 for a linear transforma- tion T V W with respect to bases a, B for V, W, respectively. know how to use the matrix representation [T5 and the coordinate map- pings R of T W to find bases for the kernel and image V, :Rm -> given two bases a, from a coordinates to 3 coordinates for Rn, know how to find the change of basis matrix
know how to find the...
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Find the matrix [T], p of the linear transformation T: V - W with respect to the bases B and C of V and W, respectively. T:P, → P, defined by T(a + bx) = b - ax, B = {1 + x, 1 – x}, C = {1, x}, v = p(x) = 4 + 2x [T] C+B = Verify the theorem below for the vector v by computing T(v) directly and using the theorem. Let V and W...
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
4. Consider the trace on M3x3 restricted to the subspace Z3x3: (Note: this is still a linear transformation, when restricted to Z3x3.) Find the matrix [trlss of this linear transformation with respect to the bases B and S. Here S is the standard basis for R. (a) What is the size of [tr]ss? (b) Compute [trss (c) Check that [tr]SBL]B--[tr(L)]s.
4. Consider the trace on M3x3 restricted to the subspace Z3x3: (Note: this is still a linear transformation, when restricted...
Consider the linear transformation T from V = P2 to W = P2 given by Tao + ayt + azt) = (-63, + 2a, + 3a2) + (2a, + 4aq + 2az)t + (220 + 2a, + 3a2)2 Let F = (F1, F2, f3) be the ordered basis in P2 given by f(t) = 1, 72(t) = 1 + t, f3(t) = 1 +t+2 Find the coordinate matrix (TFF of T relative to the ordered basis Fused in both V...
Problem 2. In each part below, either diagonalize the given linear transformation, if possible, or else explain why this is impossible. (That is, find a basis B such that the coordinate matrix [T\B or explain why no such basis exists.) (а) Т: Р2 —> Р2 given by T(p) — ар'. (b) Т:P, — P2 given by T(р) — р(2л — 1). (c) T R2x2 R2x2 given by T(A) = A+ AT. (d) T: С +С given by T(a + bi)...
For each transformation below, find the closed form of the transformation. 1) Let T be a linear transformation from R$ to M22 (R) [i Let B=1 0:00 [. :] [11] [12] [0 ] Let C= 12 41 -17 -5 65 -27 92 Let M = be the matrix transformation of T from basis B to C 17 58 -15 -51 81 The closed form of the transformation is Tb 3-1 2) Let T be a linear transformation from P3(R) to...
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