Notice that all elements of the set are letters. Moreover, they are vowels. Thus

{x|x is a vowel}

{x|x is a vowel}

Question

asked 2021-01-04

In the following question there are statements which are TRUE and statements which are FALSE.

Choose all the statements which are FALSE.

1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.

2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.

3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.

4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.

5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

Choose all the statements which are FALSE.

1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.

2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.

3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.

4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.

5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

asked 2021-03-02

Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.

asked 2021-02-09

1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices?

asked 2021-01-16

Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number \(c_1\).
\(c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}\)

asked 2021-02-05

Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:

\(\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}\)

Show that symmetric deference operation on P(A) define by the formula

\(\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}\)

(where \(\displaystyle{y}^{{c}}\) is the complement of y) the following statement istrue:

The algebraic operation o+ is commutative and associative on P(A).

\(\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}\)

Show that symmetric deference operation on P(A) define by the formula

\(\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}\)

(where \(\displaystyle{y}^{{c}}\) is the complement of y) the following statement istrue:

The algebraic operation o+ is commutative and associative on P(A).

asked 2021-02-08

Let \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) be the set of 2 x 2 matrices with the entries in \(\mathbb{Z}/\mathbb{6Z}\)

a) Can you find a matrix \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) whose determinant is non-zero and yet is not invertible?

b) Does the set of invertible matrices in \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) form a group?

a) Can you find a matrix \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) whose determinant is non-zero and yet is not invertible?

b) Does the set of invertible matrices in \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) form a group?

asked 2021-03-08

(square roots of the identity matrix) For how many 2x2 matrices A is it true that \(A^2=I\) ?
Now answer the same question for n x n matrices where n>2

asked 2021-01-17

Let V be the vector space of real 2 x 2 matrices with inner product

(A|B) = tr(B^tA).

Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for \(U^\perp\) where \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)

(A|B) = tr(B^tA).

Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for \(U^\perp\) where \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)

asked 2021-01-04

Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if

\(A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}\)

If the operation is not possible , write NOT POSSIBLE and be able to explain why

a)A+B

b)B+C

c)2A

\(A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}\)

If the operation is not possible , write NOT POSSIBLE and be able to explain why

a)A+B

b)B+C

c)2A

asked 2021-02-24

Let \(u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}\)
It can be shown that 4u-3v-w=0. Use this fact (and no row operations) to find a solution to the system Ax=b , where

\(A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}\)

\(A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}\)