Solve each of these equations using algebraic operations.

A is an m×n matrix.Check the true statements below:A. The kernel of a linear transformation is a vector space.B. If the equation

Bess [88]

Answer:

Results are (1) True. (2) False. (3) False. (4) True. (5) True. (6) True.

Step-by-step explanation:

Given A is an $m\times n$ matrix. Let $T :U\to V$ be the corresponding linear transformationover the field F and $\theta$ be identity vector in V. Now if $x\in Ker( T)\implies T(x)=\theta$ .

(1) The kernel of a linear transformation is a vector space : True.

Let $x,y\in Ker( T)$ , then,

$T(x+y)=T(x)+T(y)=\theta+\theta=\theta\impies x+y\in Ker( T)$

hence the kernel is closed under addition.

Let $\lambda\in F, x\in Ker( T)$ , then

$T(\lambda x)=\lambda T(x)=\lambda\times \theta=\theta$

$\lambda x\in Ker(T)$ and thus Ket(T) is closed under multiplication

Finally, fore all vectors $u\in U$ ,

$T(\theta)=T(\theta+(-\theta))=T(\theta)+T(-\theta)=T(\theta)-T(\theta)=\theta$

$\implies \theta\in Ker(T)$

Thus Ker(T) is a subspace.

(2) If the equation Ax=b is consistent, then Col(A) is $\mathbb R^m$ : False

if the equation Ax=b is consistent, then Col(A) must be consistent for all b.

(3) The null space of an mxn matrix is in $\mathbb R^m$

: False

The null space that is dimension of solution space of an m x n matrix is always in $\mathbb R^n$ .

(4) The column space of A is the range of the mapping $x\to Ax$

: True.

(5) Col(A) is the set of all vectors that can be written as Ax for some x. : True.

Here Ax will give a linear combination of column of A as a weights of x.

(6) The null space of A is the solution set of the equation Ax=0.

: True

5 0

3 years ago

I need help on 19 and 20

svp [43]

Number 19 you are comparing one measurement to another. Since it says 1/2 inch equals 4 ft, we want to find out how many more inches are needed if the given scale was 2/3 = 4 ft. Now lets find a common denominator for both scales stated in inches. We have 2/3 inch and 1/2 inch. Our denominator are the bottom parts of the fraction where we need to find a common factor for the denominator so we can add or subtract fractions. We have a 3 and a 2. You may always use the multiplication between two denominators to find a common factor such as 3 times 2 which equals 6 for both denominators. Now we multiplied the 3 by 2 to get 6 so the top part (numerator needs to be multiplied the the 2 because we changed the bottom part by 2 as well. You should notice that when you reduce your fraction now 4/6 is 2/3. Just a self check example there. As for 1/2 we multiplied a 3 to get 6 for the denominator so we need to multiply the numerator by 3 as well. You now should have 4/6 and 3/6. Since the question asks for how many more inches we need to subtract 4/6 from 3/6 and we get 1/6 inch for our answer.

8 0

3 years ago

What is the degree of the function f(x)=x⁴-5x+8+x²?

DENIUS [597]

First put it in decreasing power according to x.

x^4-5x+8+x^2

=x^4+x^2-5x+8

Now this question is asking what is the highest degree of x you see in this function.

4 of course.

Therefore the answer to his question is degree of 4.

Done!

7 0

3 years ago

Plz help with number 14

k0ka [10]

Answer: a = 3

Step-by-step explanation: since M is in between the two points you can look at the x and y coordinates respectively and find the difference between the the given points and then add it to the midpoint

8 0

3 years ago

A sealed-tube manometer can be used to measure pressures below atmospheric pressure. The tube above the mercury is evacuated. Wh

sp2606 [1]

The <u>correct answers</u> are:

6.5 cm = 65.00001052 torr
6.5 cm = 8665.93 Pa
6.5 cm = 0.08552632119 atm

Explanation

The conversion rate from centimeters mercury (cm Hg) to torr is
1 torr = 0.099999983814461 cm.

We have 6.5 cm; this means to convert we divide:
6.5/0.099999983814461 = 65.00001052

The conversion rate from cm Hg to Pascals (Pa) is
1 cm = 1333.22 Pa

We have 6.5 cm; this means to convert we multiply:
6.5(1333.22) = 8665.93

The conversion rate from cm Hg to atmospheres (atm) is
1 atm = 75.999995199606 cm

We have 6.5 cm; this means to convert we divide:
6.5/75.999995199606 = 0.08552632119

7 0

3 years ago

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