## Description

## Problem 1: Change of basis (2 + 2 points)

Part-1.1: Change of basis for a vector

Let � = #

�

�&, where a and b are the last two digits of your roll number. Thus, if this is

CH17B987, then � = #

8

7

&. (If your roll number ends in 00, then use a = 1 and b = 1).

1. Express this vector x in terms of new basis, �! = #

1

1

& and �” = # 1

−1

&.

Part-1.2: Change of bases for linear transformation

Recall that the problem of blending of two streams was a three-input-two-output problem.

The inputs were flowrates �!, �”, �#$% and the outputs were ℎ, �&. The gain matrix is given by:

� = # 4 2 4

0.5 1 0

&

2. How will this matrix change if the domain space is expressed in terms of the following bases:

�! = 6

1

0

0

7, �” = 6

1

1

0

7, �’ = 6

0

1

−1

7

and the co-domain space is expressed in terms of

�! = #

1

1

&, �” = # 1

−1

&

## Problem 2: Linearly Dependence (1 + 3 points)

Let �!, �”, �’ ∈ ℝ( be linearly independent vectors in n-dimensional space.

3. If the above three vectors are linearly independent, what is/are the possible value(s) of n?

(a) n = 1 (b) n = 2 (c) n = 3 (d) n = 4 (e) n = 5

4. Consider the three vectors � = �! + �”, � = �! + �’, � = �” + �’. Are the vectors u, v, w

linearly independent? Prove this.

## Problem 3: Null and Image Spaces (4 points)

Consider a matrix L = # � � 4

0.5 1 0

&, where a and b are the last two digits of your roll number.

If your roll number ends in 00, use a = 1 and b = 1.

5. Using definition, determine null space and image space of L.

Using MATLAB: Not Graded, For Practice Only

Also confirm the same using SVD (please use MATLAB for SVD).

## Problem 4: Eigenvalue Decomposition and Matrix Exponent (1 + 1 + 1 + 1 points)

Consider a matrix: � = #

1 0

� �

&, where a and b are the last two digits of your roll number.

Thus, if this is CH17B987, then � = #

1 0

8 7

&. (If your roll number ends in 00, use a = 1 and b = 1).

6. Obtain the characteristic equation and hence compute the eigenvalues of B.

7. Substitute B in its characteristic equation and thus verify Cayley Hamilton Theorem.

8. Perform eigenvalue decomposition for the matrix B

9. Using eigenvalue decomposition, compute matrix exponent �&

Using MATLAB: Not Graded, For Practice Only

Use MATLAB and compute the matrix exponent of B.

## Problem 5: Jordan Decomposition (2 + 2 points)

10. Find the eigenvalues of the matrix, � = # 1 1

−1 3

&. Since eigenvalues are repeated, compute

eigenvector and generalized eigenvector

11. Representing � = �Λ�)! in Jordan canonical form, compute the matrix exponent

Hints for Problems 4 and 5

• Please see the discussion about effect of similarity transform on exponent

• Consider the following rules

exp F#

� 0

0 �

&H = I

�* 0

0 �*J, exp F#

� �

0 �

&H = I

�* ��*

0 �* J