We have:
(a) ker(t) = {(x1, x2, x3, x4) | x1 = 3x3, x2 = -x3, x4 = t, where t is any scalar}
(b) nullity(t) = 1
(c) range(t) = span{(1, 1, -1), (-2, 5, 1), (-3, 3, 4)}
(d) rank(t) = 3
To find the kernel of the linear transformation, we need to find all vectors x such that t(x) = ax = 0. This means we need to solve the system of linear equations:
x1 - 2x2 - 3x3 = 0
x1 + 5x2 + 3x3 = 0
-x1 + x2 + 4x3 + x4 = 0
3x1 + x2 + 2x3 + x4 = 0
Putting this system into reduced row echelon form, we get:
1 0 -3 0
0 1 1 0
0 0 0 1
0 0 0 0
The pivot columns are 1, 2, and 4. So, the basic variables are x1, x2, and x4, while x3 is a free variable. So, the kernel of the linear transformation is given by:
ker(t) = {(x1, x2, x3, x4) | x1 = 3x3, x2 = -x3, x4 = t, where t is any scalar}
Therefore, the dimension of the kernel or nullity of t is 1, since there is only one free variable.
To find the range of the linear transformation, we need to find all vectors y such that y = t(x) = ax for some vector x. This is the span of the columns of the matrix A, which can be found by row reducing A to get:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0
The pivot columns are 1, 2, and 3, so the corresponding columns of A form a basis for the range of t. Therefore, the range of t is:
range(t) = span{(1, 1, -1), (-2, 5, 1), (-3, 3, 4)}
which has dimension 3. Thus, the rank of t is 3.
Therefore, we have:
(a) ker(t) = {(x1, x2, x3, x4) | x1 = 3x3, x2 = -x3, x4 = t, where t is any scalar}
(b) nullity(t) = 1
(c) range(t) = span{(1, 1, -1), (-2, 5, 1), (-3, 3, 4)}
(d) rank(t) = 3
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8(1 + 7x) - x = -40 + 7x
PLZZZ HELP!!!
Answer:
use photomath ok my guy and good job
HELP PLEASE ASAP
Given the discriminant of –2 for a quadratic equation, determine the number of real solutions.
Answer:
0 or simply no real solutions
Step-by-step explanation:
Given the discriminant which is negative, then the equation has no solutions in the set of real numbers. The roots or solutions belong to the set of complex solutions or imagenary roots.
Oliver bought 5 glow bracelets for each of his party guests and 2 glow bracelets to keep for himself. He bought a total of 37 glow bracelets. How many guests did Oliver invite to his party?
Shown is a square garden with a triangular pond.
Find the area of the garden that is grass
The area of the garden that is grass is 30 square meters.
The square garden has a side length of 6 meters.
The triangular pond has a base of 4 meters and a height of 3 meters.
The area of the garden that is grass.
The area of the square garden is 6 x 6 = 36 square meters.
The area of the triangular pond is (1/2) x 4 x 3 = 6 square meters.
Therefore, the area of the garden that is grass is 36 - 6 = 30 square meters.
The area of the garden that is grass is 30 square meters.
Here are the steps in detail:
Calculate the area of the square garden: 6 m * 6 m = 36 square meters
Calculate the area of the triangular pond: (1/2) * 4 m * 3 m = 6 square meters
Subtract the area of the pond from the area of the square to find the area of the grass: 36 square meters - 6 square meters = 30 square meters
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Pleas help if possible :)
Answer: i don’t know but good luck, you got this!Have a nice day!
Step-by-step explanation:
What must you subtract from the expression –5x – 8 in order to make it equivalent to –9x – 1 for all values of x?
Answer:
the number is 4x - 7
Step-by-step explanation:
Let P be subtracted from -5x - 8 to make it equivalent to -9x - 1.
-5x - 8 - P = -9x - 1
make "-P" the subject of the formula
-P = -9x - 1 + 5x + 8
Collect like terms;
-P = -9x + 5x - 1 + 8
-P = -4x + 7
divide through by "-1"
P = 4x - 7
Therefore, the number that must be subtracted from -5x - 8 to make it equivalent to -9x - 1 is 4x - 7
Pedro placed an order with an online nursery for
6 apple trees and 5 azaleas and the order came to
$147. The next order for 3 apple trees and 4 azaleas
came to $96. What was the unit cost for each
apple tree and for each azalea?
Hope this helps! I really need points to ask a question , so I'm just answering people who need help
Apply the distributive property to the expression 9(6m-4n). DO NOT SYMPLIFY.
Answer:
54m-36n
Step-by-step explanation:
9·6m= 54m
9·(-4n)= -36n
54m-36n
Locate the opposite of the number 1 5/8 on this number line. Draw a dart on correct position of the number line
Answer:
-1 5/8
Step-by-step explanation:
Opposite of 1 5/8 is -1 5/8
use the distributive property to rewrite the algebraic expression below
4 (x + 2)
2/5 of 146 as a decimals
Frankie started cleaning her house at 1:30 p.M and finished cleaning at 6;30 p.M how long did Frankie clean
Answer:
5 hours
Step-by-step explanation:
Start time = 1:30 p.m
Finishing time = 6:30 p.m
Time taken :
Finishing time - start time
6:30 p.m - 1.30 p.m
= 5 hours
next three terms in (1024, -128, 16...) you can write them as fraction if needed
Answer:
-2 1/4 - 1/32
Step-by-step explanation:
16x -1/8 = -2 do the same for all
What is the base of the expression 11/12
Answer:
11
Step-by-step explanation:
Answer:
11
Step-by-step explanation:
Please help. I don’t know what to do.
Answer:
83
Step-by-step explanation:
The sum of the interior angles of any closed figure will always equal 360º, so what you need to do is add:
109+90+78=?
This will in turn give you 277.
Subtract 277 from 360, and boom, 83 degrees.
PLEASE HELP ME ASAP.
Answer:
use pathway but with an M instead of P
Step-by-step explanation:
An article marked at ₹50 is sold for ₹45. What is the
rate of discount?
Answer:
Your answer is
Rate of discount = 10%
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10÷2-3+1=3 where do the parentheses go?
Answer:
The parantheses go around the (10 / 2)
Step-by-step explanation:
The expression would be:
(10 / 2) - 3 + 1 = 3
And when you solve it,
It would simplify to:
5 - 3 + 1
go from left to right and you get 3
The slash means division, I don't know how to type a division sign on brainly lol
You are schools you are a school supply shopping in together as a package of six pink pencils for 250 how much does each pencil cost
Answer:
41.67 dollars
Step-by-step explanation:
because 250/6= 41.6666666667
that rounded is 41.67 dollars
to complete the equation and
Use the graph to determine which values can be used
x^2/a^2 - y^2/b^2= 1.
a:
b:
Answer:
/9AZ+ 1AL+4UE, 1, 5Z0 971 349 BG, -, 5Z0 971 349 BG -. (1). PR-9AA/9AZ+ 1AC+4UA. PR-9AA/9AZ+ 1AC+4UA, 1, 5Z0 971 349 AR, -, 5Z0 971 349 AR
Step-by-step explanation:
Answer:
A:5
B:12
Step-by-step explanation:
The ratio of small dogs to large dogs at a show is 4:3. If there are a total of 56 dogs in the show, how many are large dogs?
A triangle has side lengths of 11 cm and 7cm. Which could be the value of the third side, 19cm or 17 cm
Answer:
use the Pythagoras theorem..you will got the solution
Answer:
The answer is 17
Step-by-step explanation:
write the fraction in ascending order.
1) 2 , 5 , 1 , 3
----- ----- ----- ----
3 6 2 4
2) 4 , 5 , 3 , 1
---- ----- --- ---
5 8 4 2
QUESTION 1 :
find the LCM of 3,6,2,4
LCM = 12
THEREFORE,
2/3*4/4 = 8/12
5/6*2/2 = 10/12
1/2*6/6 = 6/12
3/4*3/3 = 9/12
8/12 , 10/12 , 6/12 , 9/12
Ascending order of ( 8/12 , 10/12 , 6/12 , 9/12)
= 6/12 , 8/12 , 9/12 , 10/12
So, 6/12 = 1/2
8/12 = 2/3
9/12 = 3/4
10/12 = 5/6
THEREFORE
Ascending order = 1/2 , 2/3 , 3/4 , 5/6
84,000 of food for 80 soldiers for 60 days How long would the food last 20 more soldiers joined
Answer:
If 20 more soldiers join it would last 48 days
Step-by-step explanation:
if 80 soldiers need 84000 food for 60 days it means that in 60 days one soldier eats 84000/80= 1050 each
then 1050/60=17.5 meaning they need 17.5 food a day each,
adding 20 soldiers means 80+20= 100 soldiers
100*17.5=1750 food is eaten every day
84000/1750= 48 which means that 100 soldiers will eat 84000 in 48 days
HELP IT WOULD MEAN A LOT IT'S DUE TODAY IN 7 MINUTES
Answer:
39
Step-by-step explanation:
39 + 32 = 71
hope this helps have a good day!!
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What is the y-intercept of the line in the picture
Answer:
-2
Step-by-step explanation:
write a polynomial function with zeros -3,5,8
Answer:
x^3-10x^2+x+120
Step-by-step explanation:
Assuming you mean roots -3, 5, 8
These happen when we have (x+3)(x-5)(x-8)=0
Expand this
(x^2-2x-15)(x-8)=0
=x^3-2x^2-15x-8x^2+16x+120
=x^3-10x^2+x+120
Corresponding angles and interior angles can never have equal angle
True or false
Plss help mee
Answer:
true
Step-by-step explanation:
5 circles lie on the same plane.What is the maximum number of intersection points the circles can have?
the answer is 10 pts bc the maximum number of points of intersection when 5 lines are drawn in a plane, as shown, is 10 points.
Answer:20
Step-by-step explanation: The rule you use for the problems is the same you use for finding the sum of patterned numbers: n*(n+1). n is equivalent to the number of circles. Thus, 5 *(5-1)= 5*4=20
Hope it helps!
Help!!!! What are the unknown angles?
Answer:
1: 52 for x
2. 96 for x
3. 58 for x
4. 71 for both of the 2 angles, 71 each
Step-by-step explanation:
all triangles add up to 180*, easy to reverse the math
uwu