2. Let \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2\}\) be a basis for vector space \(V\). Let \(T : V \to V\) be a linear transformation with the property that\[T(\mathbf{b}_1) = 2\mathbf{b}_1 - 3\mathbf{b}_2, \quad T(\mathbf{b}_2) = -4\mathbf{b}_1 + 5\mathbf{b}_2\]Find \([T]_{\mathcal{B}}\), the matrix for \(T\) relative to \(\mathcal{B}\).

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Answered: 2. Let B = {b₁, b2} be a basis for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
B = {bj, ..., b, } be a basis for a vector space V. Explain why the B-coordinate vectors of b1, ..., b, are the columns e1, en of the n x n identity matrix.
Let B = {b, b, b3} be a basis for vector space V. Let T: V → V be a linear transformation with the following properties. T(b,) = - 8b, - 6b2: T(b2) = - 2b, +4b2: T(b3) = - 7b, %3D Find [T]B, the matrix for T relative to B.
B = (v1, v2, V3) is a basis of the vector space V and T :V →V is a linear transformation which satisfies T(vı) = v1 + v2 + 2v3, T(v2) = 2v1 + v2 + 3v3, T(v3) = vị + 2v2 + 3v3. If v = vị – v2 + 2v3 then T(v) = a) vi + 4v2 + 5v3 b) vị – v2 + 2v3 c) 6v1 + 6v2 + 9v3 d) 4v1 + 7v2 +5v3 e) 5v1 + 6v2 + 1lv3
Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.
B = (v1, v2, v3) is a basis of a vector space V andT :V → V is a linear transformation which satisfies T(v1) = v1 + v2 + 2v3,T(v2) = 2v1 + v2 + 3v3, T(v3) = v1 + 2v2 + 4v3. If u = 4v1 – v2 – vz and T(u) = c1v1 + c2v2 + c3V3 then what is the value of cı + c2 + c3? - O 4 O 5 O 3 O 7 2.
Consider the linear transformation T: C→C of the last problem. Find the matrix T that represents it in each of the following basis: a) {1, i}, b) {1+i, 1+2i}.

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2. Let B = {b₁, b2} be a basis for vector space V. Let T: V → V be a linear transformation with the property that T(b₁) = 2b₁ – 3b2, T(b2) = −4b₁ + 5b2 Find [T], the matrix for T relative to B.