3. For V C R" and W C R"m, assume that V, W are vector subspaces and B := {v1, v2, .., v,} and C := {w1, W2, .., Ws} are bases for V and W respectively. Let T : V → W be a linear transformation and A:= (a;;) be an (s x r)-matrix such that T(v1) = a11Wı+421W2 + · · · + asıWs T(v2) = a12Wı+a22W2+ • · ·+ as2Ws T(v,) = a1,Wi+a2,W2 + ·..+asr Ws. Show that for all u e V, [T(u)]c = A·[u]g. Hint: For i = 1.r, find [T(v;)]c and then, write u = ¤1V1 + X2V2 ++x,Vr € V. %3D
3. For V C R" and W C R"m, assume that V, W are vector subspaces and B := {v1, v2, .., v,} and C := {w1, W2, .., Ws} are bases for V and W respectively. Let T : V → W be a linear transformation and A:= (a;;) be an (s x r)-matrix such that T(v1) = a11Wı+421W2 + · · · + asıWs T(v2) = a12Wı+a22W2+ • · ·+ as2Ws T(v,) = a1,Wi+a2,W2 + ·..+asr Ws. Show that for all u e V, [T(u)]c = A·[u]g. Hint: For i = 1.r, find [T(v;)]c and then, write u = ¤1V1 + X2V2 ++x,Vr € V. %3D
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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please help with question 3
![3. For V C R" and W C R", assume that V, W are vector subspaces and B := {v1, V2, .., V,}
and C := {w1, w2, .., Ws} are bases for V and W respectively. Let T: V → W be a linear
transformation and A := (a;;) be an (s × r)-matrix such that
T(v1) = a11W1+A21W2 +.+ as1Ws
T(v2) = a12Wi+A22W2 +
... + as2Ws
T(vr) = a1,W1+a2„W2 +..+asr Wg.
Show that for all u e V, [T(u)]c = A·[u]g.
Hint: For i = 1..r, find [T(vi)]c and then, write u = *1V1 + x2V2 + · · · + x,Vr € V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F647d1c74-04a7-437b-8d81-5635d9764604%2F09b202fc-1db3-4b10-a27c-6ac55602a7bc%2Fnbrrmdk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. For V C R" and W C R", assume that V, W are vector subspaces and B := {v1, V2, .., V,}
and C := {w1, w2, .., Ws} are bases for V and W respectively. Let T: V → W be a linear
transformation and A := (a;;) be an (s × r)-matrix such that
T(v1) = a11W1+A21W2 +.+ as1Ws
T(v2) = a12Wi+A22W2 +
... + as2Ws
T(vr) = a1,W1+a2„W2 +..+asr Wg.
Show that for all u e V, [T(u)]c = A·[u]g.
Hint: For i = 1..r, find [T(vi)]c and then, write u = *1V1 + x2V2 + · · · + x,Vr € V.
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