-8 1 V₁ = These vectors form a basis for W, the subspace of R4 that we encountered in Activity 6.4.2. Since these vectors are the columns of A, we have Col(A) = W. W1 = V1 W2 = V2 Evaluate (Sage) a. When we implemented Gram-Schmidt, we first found an orthogonal basis w₁. w2, and w3 using W3 = V3 V2. W1 W₁-W1 V3 W1 W1-W1 V3 W1 -3 3 WI V3 - W2 W2- W2 - W2 Use these expressions to write V₁, V₁, and v3 as linear combinations of w₁. w2, and w3. b. We next normalized the orthogonal basis w₁, W₂, and w3 to obtain an orthonormal basis u₁, 2, and u3. Write the vectors w, as scalar multiples of u. Then use these expressions to write v₁. v₁, and v3 as linear combinations of u₁. 112. and 13. c. Suppose that Q = [u₁ 12 13]. Use the result of the previous part to find a vector r₁ so that Qr₁ = v₁.
-8 1 V₁ = These vectors form a basis for W, the subspace of R4 that we encountered in Activity 6.4.2. Since these vectors are the columns of A, we have Col(A) = W. W1 = V1 W2 = V2 Evaluate (Sage) a. When we implemented Gram-Schmidt, we first found an orthogonal basis w₁. w2, and w3 using W3 = V3 V2. W1 W₁-W1 V3 W1 W1-W1 V3 W1 -3 3 WI V3 - W2 W2- W2 - W2 Use these expressions to write V₁, V₁, and v3 as linear combinations of w₁. w2, and w3. b. We next normalized the orthogonal basis w₁, W₂, and w3 to obtain an orthonormal basis u₁, 2, and u3. Write the vectors w, as scalar multiples of u. Then use these expressions to write v₁. v₁, and v3 as linear combinations of u₁. 112. and 13. c. Suppose that Q = [u₁ 12 13]. Use the result of the previous part to find a vector r₁ so that Qr₁ = v₁.
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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![Activity 6.4.4. Suppose that A is the 4 x 3 matrix whose columns are
-B
VI
1
V2
These vectors form a basis for W, the subspace of R4 that we encountered in
Activity 6.4.2. Since these vectors are the columns of A, we have Col(A) = W.
✔
W1 = V1
W2 = V2
W3 = V3
Evaluate (Sage)
a. When we implemented Gram-Schmidt, we first found an orthogonal basis w₁.
w2, and w3 using
V2 - W1
W1-W1
V3 - W1
W1-W1
V3
W1
WI
-3
V3 - W2
W2 - W2
W2.
Use these expressions to write v₁, v₁, and v3 as linear combinations of w₁.
w2, and w3.
b. We next normalized the orthogonal basis w₁, W2, and w3 to obtain an
orthonormal basis u₁, u₂, and u3.
Write the vectors w; as scalar multiples of u,. Then use these expressions to
write v₁, v₁, and v3 as linear combinations of u₁, ₂, and 13.
c. Suppose that Q = [u₁ 1₂ 13]. Use the result of the previous part to find a
vector r₁ so that Qr₁ = V₁.
d. Then find vectors r2 and r3 such that Qr₂ = v₂ and Qr3 = V3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a7e99bb-b9c4-47fa-9081-3b9aeb6b94e4%2Fb11071fc-e3a6-41de-ba6f-19c466104650%2F17eucf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Activity 6.4.4. Suppose that A is the 4 x 3 matrix whose columns are
-B
VI
1
V2
These vectors form a basis for W, the subspace of R4 that we encountered in
Activity 6.4.2. Since these vectors are the columns of A, we have Col(A) = W.
✔
W1 = V1
W2 = V2
W3 = V3
Evaluate (Sage)
a. When we implemented Gram-Schmidt, we first found an orthogonal basis w₁.
w2, and w3 using
V2 - W1
W1-W1
V3 - W1
W1-W1
V3
W1
WI
-3
V3 - W2
W2 - W2
W2.
Use these expressions to write v₁, v₁, and v3 as linear combinations of w₁.
w2, and w3.
b. We next normalized the orthogonal basis w₁, W2, and w3 to obtain an
orthonormal basis u₁, u₂, and u3.
Write the vectors w; as scalar multiples of u,. Then use these expressions to
write v₁, v₁, and v3 as linear combinations of u₁, ₂, and 13.
c. Suppose that Q = [u₁ 1₂ 13]. Use the result of the previous part to find a
vector r₁ so that Qr₁ = V₁.
d. Then find vectors r2 and r3 such that Qr₂ = v₂ and Qr3 = V3.
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