Question

The following is a linear algebra problem for inner product, length, and organality: Let C = {e1, e2, e3, e4} be the canonical basis of R4. Let B = {u1, u2, u3, u4} be an orthogonal basis for R4 where u1 = [0; 1; -4; -1], u2 = [3; 5; 1; 1], u3 = [1; 0; 1; -4], u4 = [5; -3; -1; 1]. Write each ei as a linear combination of the vectors in B using the dot product. Next What is the matrix of change of basis from C to B

Answer 1

The matrix of change of basis from the canonical basis C to the basis B, spanned by u₁, u₂, u₃, and u₄, is:

`\[ [I]_B^C = \begin{bmatrix} 0 & 3 & 1 & 5 \\ 3 & -1 & 4 & -1 \\ -4 & 5 & -4 & -3 \\ -1 & 4 & -4 & 1 \end{bmatrix} \]`

The question asks you to find the matrix of change of basis from the canonical basis C to the basis B, which is spanned by the vectors u₁, u₂, u₃, and u₄.

To find the matrix of change of basis, you can do the following:

1. Express each vector in the canonical basis C as a linear combination of the vectors in B using the dot product. This will give you the columns of the matrix of change of basis.

2. Once you have the columns of the matrix, you can write it down explicitly.

Here are the steps to solve the problem:

(iii) Expressing the vectors in the canonical basis C as a linear combination of the vectors in B:

We can use the dot product to express each vector in the canonical basis C as a linear combination of the vectors in B. For example, to express e₁ as a linear combination of u₁, u₂, u₃, and u₄, we can take the dot product of e₁ with each of the vectors in B:

e₁ ⋅ u₁ = (1, 0, 0, 0) ⋅ (0, 1, -4, -1) = 0

e₁ ⋅ u₂ = (1, 0, 0, 0) ⋅ (3, 5, 1, 1) = 3

e₁ ⋅ u₃ = (1, 0, 0, 0) ⋅ (1, 0, 1, -4) = 1

e₁ ⋅ u₄ = (1, 0, 0, 0) ⋅ (5, -3, -1, 1) = 5

Therefore, we can express e₁ as a linear combination of the vectors in B as follows:

e₁ = 3u₂ + u₃ + 5u₄

Similarly, we can express the other vectors in the canonical basis C as linear combinations of the vectors in B:

e₂ = -u₁ - 5u₃ + 4u₄

e₃ = 4u₁ + u₂ - 4u₄

e₄ = -u₁ - 3u₂ + u₃

(iv) Finding the matrix of change of basis from C to B:

Now that we have expressed each vector in the canonical basis C as a linear combination of the vectors in B, we can write down the matrix of change of basis. The matrix will have the following form:

[I]_B^C =

[ [u₁_B] [u₂_B] [u₃_B] [u₄_B] ]

[ [e₁_B] [e₂_B] [e₃_B] [e₄_B] ]

where uᵢ_B represents the vector uᵢ expressed in the basis B, and eⱼ_B represents the vector eⱼ expressed in the basis B.

Using the results from part (iii), we can fill in the matrix as follows:

[I]_B^C =

[ [0] [3] [1] [5] ]

[ [3] [-1] [4] [-1] ]

[ [-4] [5] [-4] [-3] ]

[ [-1] [4] [-4] [1] ]

Therefore, the matrix of change of basis from C to B is:

[I]_B^C =

[ [0 3 1 5] ]

[ [3 -1 4 -1] ]

[ [-4 5 -4 -3] ]

[ [-1 4 -4 1] ]