Matrix Algebra and Linear Transformations

Question 1

What is a zero matrix?

Answer 1

A matrix whose entries are all zero.

Question 2

LINEAR COMBINATION OF VECTORS AS THE PRODUCT OF A MATRIX AND A VECTOR

it is a fundamental idea to see the linear combination like this.

Answer 2

A x = 
                       |x1|
| a1 a2 ... an|  |x2| = x1 a1 + x2 a2 + ... + xn an
                      | ... | 
                      |xn|

Note that A x is defined only if the number of columns of A equals the number of entries in x.

!!!This is completely off, i dont know why they distort my text

Question 3

A system of linear equations can be viewed in three different but equivalent ways. Which ones?

Answer 3

1. as a matrix equation Ax = b
2. as a vector equation (x1 a1 + … + xn an = b)
3. as an augmented matrix of a system of linear equations [a1 a2 … a3] = b

Question 4

In the context of the matrix equation Ax = b. The equation has a solution if and only if …

Answer 4

… b is a linear combination of the columns of A.

Question 5

Just for your knowledge now:

Answer 5

A - m x n matrix. the following statements are logically equivalent. Either all of them are true, or all false:

1. For each b in Rˆm, the equation Ax = b has a solution
2. Each b in Rˆm is a linear combination of the columns of A
3. The columns of A span (generate) Rˆm
4. A has a pivot position in every row.

This is about the coefficient matrix, not the augmented one.

Question 6

What is an identity matrix?

Answer 6

The identity matrix is a square matrix that has 1s on the main diagonal and 0 as the rest of the matrix values.

Question 7

A transformation (or mapping) T is linear if:

Answer 7

i. T(u+v) = T(u) + T(v), for all u, v in the domain of T

ii. T(cu) = cT(u) for all scalars c and all u in the domain of T

Question 8

Every matrix transformation is a linear transformation. True or false?

Answer 8

True

Question 9

Name two applications of the dot-product of the matrix-vector multiplication?

Answer 9

Downsampling

Blurring

Question 10

You can multiply one m x n matrix by a n x p matrix in the following way:

Answer 10

You separate the second matrix by columns and you multiply each column with the first matrix.

After doing that with all the columns of the second matrix, you put them side by side in order and you get the result.

Question 11

If you multiply a 3 x 5 matrix and a 5 x 2 matrix, what is the size of the resulting matrix?

Answer 11

3 x 2

Question 12

Do you know what the row-column rule for multiplying two matrices is?

Answer 12

You take the first row of the first matrix and multiply it by the first column of the second matrix. Then you have the value for the position 11 in the resulting matrix.

You take the first row of the first matrix and multiply it by the second column of the second matrix. Then you have the value for the position 12 in the resulting matrix.

And so on

Question 13

Name some properties of matrix multiplication

Answer 13

associative law, distributive law, identity matrix multiplication

Question 14

A matrix A is said to be invertible if there is another matrix C that has the same dimensions such that …

!!! only when there are square matrices

Answer 14

CA = I and AC = I

Question 15

What is a singular matrix?

Answer 15

A non-invertible one.

Question 16

A matrix A is invertible id its determinant is …

Answer 16

not equal to 0

Question 17

How do you calculate the determinant of a 2 x 2 matrix

Answer 17

ad - cb

Question 18

When you know that a 2 x 2 is invertible how do you calculate the inverse?

Answer 18

1/detA [ d -b

-c a]

Question 19

If A is an invertible n x n matrix, then for each b in Rˆn, the equation Ax = b has the unique solution …

Answer 19

x = Aˆ(-1) b

Question 20

What is the transpose of a matrix?

Answer 20

A matrix that has interchanged the values on the rows and the values on the column. The dimensions change as well?

Question 21

What is the inverse of the inverse of a matrix?

Answer 21

The matrix itself.

Question 22

Do you know the algorithm of finding the inverse of a matrix by appending I to it?

Answer 22

You append I to the original matrix and you do row reduction to reduced echelon form only focusing on the original matrix, and then you will see that I has changed to the inverse of the matrix.

Question 23

It would be a good idea to check what matrix partitioning is.

Answer 23

A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.

Check more on the slides about it.