2.Let {v1, v2, v3}, {w1, w2, w3} be two sets of linearly independent vectorsin R3.(1)Show th it there exists a 3 × 3 matrix A such thatAvi =w1,Av2 =w2,Av3 =w3.(2)Let B be any 3 x 3 matrix such thatBvi =w1;Bv2 =w2,Bv3 =w3.Show that A = B. In other words, the matrix A constructed in (1) is unique.

Here we will prove the required result.

2. Let {v1, v2, V3}, {w1, w2, w3} be two sets of linearly independent vectors in R³. (1) Show th it there exists a 3 × 3 matrix A such that Avı =w1, Av2 =w2, Av3 =w3. (2) Let B be any 3 x 3 matrix such that Bvi =w1, Bv2 =w2, Bv3 =w3. Show that A = B. In other words, the matrix A constructed in (1) is unique.

2. Let {v1, v2, V3}, {w1, w2, w3} be two sets of linearly independent vectors in R³. (1) Show th it there exists a 3 × 3 matrix A such that Avı =w1, Av2 =w2, Av3 =w3. (2) Let B be any 3 x 3 matrix such that Bvi =w1, Bv2 =w2, Bv3 =w3. Show that A = B. In other words, the matrix A constructed in (1) is unique.

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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2.
Let {v1, v2, v3}, {w1, w2, w3} be two sets of linearly independent vectors
in R3.
(1)
Show th it there exists a 3 × 3 matrix A such that
Avi =w1,
Av2 =w2,
Av3 =w3.
(2)
Let B be any 3 x 3 matrix such that
Bvi =w1;
Bv2 =w2,
Bv3 =w3.
Show that A = B. In other words, the matrix A constructed in (1) is unique.

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