MTH501 Final Term Questions Solution

To determine the rank of the matrix 𝐴 given its order and the dimension of its null space, we can use the Rank-Nullity Theorem.

where:

In this problem:

The null space of 𝐴 is two-dimensional, so nullity(𝐴) = 2. Using the Rank-Nullity Theorem:

rank(𝐴) + 2 = 11

Solving for rank(𝐴):

rank(𝐴) = 11 − 2

rank(𝐴) = 9

So, the rank of the matrix 𝐴 is 9. To solve the problem of finding the matrix representation of the linear transformation

1. Identify the images of the basis elements under 𝑇. 2. Express these images as linear combinations of the basis elements. 3. Form the matrix from these linear combinations.

1. 2. 3.

1. 2. 3.

Step 3: Form the matrix

The matrix representation of 𝑇 relative to the basis 𝐵 is formed by taking the coefficients of these linear combinations and arranging them into columns. Specifically, the matrix

−2 −4 −1

−2 −4 −1 To compute the least square error associated with the least squares solution 𝑥ˆ of the

1. column space of 𝐴. 2. 3. Calculate the least square error, which is the norm of the residual vector squared.

Given:

𝐴 = −2 −1 , 𝑏 = −4 − 1

−2 −1 − 1 1 ⋅ 4 3 + ( − 2) ⋅− 1

−1 ⋅ 4 3 + 2 ⋅− 1

0 ⋅ 4 3 + 3 ⋅− 1

2 ⋅ 4 3 + 5 ⋅− 1 3 + 2

− 4 3 − 2

−1 3 − 5 −2 −1

Step 2: Compute the residual vector 𝑟

The residual vector 𝑟 is given by: