Practice Problems

1. Which of the following is true for any matrix A?

NOTE: dim(C(A)) ==> dimension of column space of A, dim(N(A)) ==> dimension of null space of A

rank(A) > dim(C(A))
dim(C(A)) = dim(N(A))
dim(N(A)) + dim(C(A)) = dimension of whole vector space
None

View Answer

> _dim(N(A)) + dim(C(A)) = dimension of whole vector space_

2. Let A, B, C, D be n × n matrices. If ABCD = 1, then B^-1?

D^-1C^-1A^-1
CDA
ADC
None
Insufficient information

View Answer

> _Insufficient information. Since it is not given that the matrices are invertible. If invertible the answer would be ==> CDA_

3. Find a transformation matrix in 3D space, that first rotates the space about X axis by 270^o, then rotates the resulting space about its Y axis by 90^o and finally rotates the resulting space about its Z axis by 180^o (All rotations in clockwise direction)

View Answer

Let A ==> Rotation about X by -270^o B ==> Rotation about Y by -90^o C ==> Rotation about Z by -180^o since clockwise* A = | 1 | 0 | 0 | | :---: | :---: | :---: | | 0 | 0 | 1 | | 0 | -1 | 0 | B = | 0 | 0 | -1 | | :---: | :---: | :---: | | 0 | 1 | 0 | | 1 | 0 | 0 | C = | -1 | 0 | 0 | | :---: | :---: | :---: | | 0 | -1 | 0 | | 0 | 0 | 1 | Resultant transformation = **C x B x A** Think why preorder multiplication?

4. Does all natural numbers in range (0, ∞) form a vector space or sub space? Justify

View Answer

> _No, Since zero vector is not present_

5. Does all natural numbers in range (-∞, 0] form a vector space or sub space? Justify

View Answer

> _For any space to be a vector space/subspace, there should be closure i.e resultant vector after linear transformation should lie in same space. But here the resultant can be positive and hence the answer is NO_

6. Does all 2D matrices form a vector space? Justify

View Answer

> _Yes_

7. A transformation matrix squishes 4D space into line, what is the rank of that matrix?

View Answer

> _1 (Since line)_

8. How many basis exists for a 4D vector space and how many vectors are there in that basis

View Answer

> _Infinite number of basis, each basis consists of 4 vectors_

9. Prove that determinant of the matrix given below is , det(A) = ad - bc


a	b
c	d

View Answer

![det-proof](/linear-algebra-study-group/001-3b1b_Revision/Images/Det_Proof.png) Area of resultant parallelogram = (a+b)x(c+d) - [2bc + ac/2 + ac/2 + bd/2 + bd/2]

Practice Problems

1. Which of the following is true for any matrix A?

2. Let A, B, C, D be n × n matrices. If ABCD = 1, then B-1?

3. Find a transformation matrix in 3D space, that first rotates the space about X axis by 270o, then rotates the resulting space about its Y axis by 90o and finally rotates the resulting space about its Z axis by 180o (All rotations in clockwise direction)

4. Does all natural numbers in range (0, ∞) form a vector space or sub space? Justify

5. Does all natural numbers in range (-∞, 0] form a vector space or sub space? Justify

6. Does all 2D matrices form a vector space? Justify

7. A transformation matrix squishes 4D space into line, what is the rank of that matrix?

8. How many basis exists for a 4D vector space and how many vectors are there in that basis

9. Prove that determinant of the matrix given below is , det(A) = ad - bc

2. Let A, B, C, D be n × n matrices. If ABCD = 1, then B^-1?

3. Find a transformation matrix in 3D space, that first rotates the space about X axis by 270^o, then rotates the resulting space about its Y axis by 90^o and finally rotates the resulting space about its Z axis by 180^o (All rotations in clockwise direction)