Questions on Algebra: Linear Algebra (NOT Linear Equations) answered by real tutors!

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Question 445381: What are the solutions to
6x+3y+z=19
x-3y+2z=21
17x-2y+3z=86
Click here to see answer by MathTherapy(10586)

Question 445381: What are the solutions to
6x+3y+z=19
x-3y+2z=21
17x-2y+3z=86
Click here to see answer by ikleyn(53419)

Question 1165604: Consider the following functions T : R → R . Show that each is a linear transformation and determine for each the matrix A such that T (x) = A x
a) x1. x1+2x2 +3x3
T x2. = 2x2-3x1+x3
¶ x3
b) x1. 7x1+2x2 +x3
T x2. = 3x1-11x2+2x3
¶ x3
Click here to see answer by CPhill(2138)

Question 1165858: Let matrix3x2 A = 1, -1 , 2, 0, 3, -4. Determine whether multiplication by A is a one-to-one transformation.
Click here to see answer by CPhill(2138)

Question 1165864: Kuttler, pg. 279, Ex. 5.2.8) Consider the following functions T : R3→ R2 . Show that each is a linear transformation and determine for each the matrix A such that
T (x) = A x.
T . (matrix3x1(x1, x2, x3) )= Matrix2x1(x1+2x2+3x3, 2x2−3x1+x3)
Click here to see answer by CPhill(2138)

Question 1166341: Let W be the set of all sequences which converge to 0. Is W a subspace of the vector space of convergent sequences? You must justify your answer.
Click here to see answer by CPhill(2138)

Question 1166343: 6 Let V = C[a, b] be the vector space of continuous functions on interval
[a, b]. Let W be the subset of functions in V such that:
(integralsign) from(a,b) f(x) dx = 0.
Is W a subspace of V ? You must justify your answer.
Click here to see answer by CPhill(2138)

Question 1166345: Yes, **W**, the set of skew-symmetric $n \times n$ matrices, is a **subspace** of $\mathbf{V}$, the vector space of all $n \times n$ matrices.
To prove that $W$ is a subspace of $V$, we must verify the three conditions for a subset to be a subspace:
1. **The Zero Vector Condition:** The zero vector of $V$ must be in $W$.
2. **Closure under Vector Addition:** If $\mathbf{A}$ and $\mathbf{B}$ are in $W$, then their sum $(\mathbf{A} + \mathbf{B})$ must also be in $W$.
3. **Closure under Scalar Multiplication:** If $\mathbf{A}$ is in $W$ and $c$ is any scalar, then the scalar multiple $(c\mathbf{A})$ must also be in $W$.
Recall that a matrix $\mathbf{A}$ is **skew-symmetric** if and only if $\mathbf{A}^T = -\mathbf{A}$.
***
### 1. The Zero Vector Condition
The zero vector in $V$ is the $n \times n$ **zero matrix**, $\mathbf{Z}$, where every entry is $0$.
We check if $\mathbf{Z}$ is skew-symmetric:
$$\mathbf{Z}^T = \mathbf{Z}$$
$$-\mathbf{Z} = -\mathbf{Z} \text{ (which is still } \mathbf{Z})$$
Since $\mathbf{Z}^T = \mathbf{Z} = -\mathbf{Z}$, the condition $\mathbf{Z}^T = -\mathbf{Z}$ holds.
Thus, the zero matrix $\mathbf{Z}$ is skew-symmetric, and $\mathbf{Z} \in W$.
### 2. Closure under Vector Addition
Let $\mathbf{A}$ and $\mathbf{B}$ be two arbitrary matrices in $W$. This means:
* $\mathbf{A}^T = -\mathbf{A}$
* $\mathbf{B}^T = -\mathbf{B}$
We need to check if their sum $(\mathbf{A} + \mathbf{B})$ is also skew-symmetric. We check the transpose of the sum:
$$(\mathbf{A} + \mathbf{B})^T = \mathbf{A}^T + \mathbf{B}^T \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric conditions:
$$(\mathbf{A} + \mathbf{B})^T = (-\mathbf{A}) + (-\mathbf{B})$$
$$(\mathbf{A} + \mathbf{B})^T = -(\mathbf{A} + \mathbf{B})$$
Since the transpose of $(\mathbf{A} + \mathbf{B})$ is equal to $-(\mathbf{A} + \mathbf{B})$, the sum $(\mathbf{A} + \mathbf{B})$ is skew-symmetric.
Thus, $W$ is closed under matrix addition.
### 3. Closure under Scalar Multiplication
Let $\mathbf{A}$ be an arbitrary matrix in $W$ (so $\mathbf{A}^T = -\mathbf{A}$), and let $c$ be an arbitrary scalar.
We need to check if the scalar multiple $(c\mathbf{A})$ is also skew-symmetric. We check the transpose of the multiple:
$$(c\mathbf{A})^T = c(\mathbf{A}^T) \quad (\text{Property of Transpose})$$
Substitute the skew-symmetric condition for $\mathbf{A}$:
$$(c\mathbf{A})^T = c(-\mathbf{A})$$
$$(c\mathbf{A})^T = -(c\mathbf{A})$$
Since the transpose of $(c\mathbf{A})$ is equal to $-(c\mathbf{A})$, the scalar multiple $(c\mathbf{A})$ is skew-symmetric.
Thus, $W$ is closed under scalar multiplication.
***
Since $W$ satisfies all three conditions, it is a **subspace** of the vector space $V$.
---
Would you like to know the **dimension** of the subspace $W$ of skew-symmetric $n \times n$ matrices?
Click here to see answer by CPhill(2138)

Question 1166667: : Let V be the vector space of all polynomials defined on the real number line. (12 points) Suppose that T : V → V is the transformation T(p(x)) = xp′(x) where the prime denotes
derivative.
(a) Show T is a linear transformation. (b) Show T is not one-to-one.
Click here to see answer by CPhill(2138)

Question 736146: A basketball player scored 18 times during one game. He scored a total of 30 points, which consists of 1 point free throws and 2 point baskets. How many one point free throws did he make? How many 2 point baskets did he make?
Click here to see answer by mccravyedwin(417)

Question 629298: Find the quadratic polynomial whose graph goes through the points (−1,5), (0,5), and (2,29).
f(x)= _x2+_ x+_ <<< that form

I got to the step 4a+2b+0=29
that was like my 6/7th step. I am getting stuck from there. Can i get any help please?

Click here to see answer by Edwin McCravy(20077)

Question 1167408: Giovanni invested N$90 000 in three different account at the beginning of 2018 according to the
following table.
2018 yield
Saving 6%
Unit trust 7%
32 days 5%
If he invested the same amount in the unit trust as well as in the 32 days accounts and his 2018 yield
for the year from the saving and the 32 days amounted to N$400, how much did he invest in each
account? Use Gaussian elimination.
Click here to see answer by CPhill(2138)

Question 1202927: CalJuice Company has decided to introduce three fruit juices made from blending two or more concentrates. These juices will be packaged in 2-qt (64-oz) cartons. One carton of pineapple-orange juice requires 8 oz each of pineapple and orange juice concentrates. One carton of orange-banana juice requires 12 oz of orange juice concentrate and 4 oz of banana pulp concentrate. Finally, one carton of pineapple-orange-banana juice requires 4 oz of pineapple juice concentrate, 8 oz of orange juice concentrate, and 4 oz of banana pulp concentrate. The company has decided to allot 16,000 oz of pineapple juice concentrate, 24,000 oz of orange juice concentrate, and 5000 oz of banana pulp concentrate for the initial production run. The company also stipulated that the production of pineapple-orange-banana juice should not exceed 800 cartons. Its profit on one carton of pineapple-orange juice is $1.00, its profit on one carton of orange-banana juice is $0.80, and its profit on one carton of pineapple-orange-banana juice is $0.90. To realize a maximum profit, how many cartons of each blend should the company produce?
pineapple-orange juice ____ cartons
orange-banana juice ___ cartons
pineapple-orange-banana juice 800 cartons
What is the largest profit it can realize?
$ ______

Are there any concentrates left over? (If so, enter the amount remaining. If not, enter 0.)
pineapple concentrate
0 oz
orange concentrate
0 oz
banana pulp concentrate
____ oz

Click here to see answer by ikleyn(53419)

Question 1160968: Let L be the line spanned by [-1, 4, 9, 0] in R^4
Find a basis of the orthogonal complement L⊥ of L.

Click here to see answer by ikleyn(53419)

Question 1167649: Suppose u, v ∈ R3. Determine if the function
<> = 2u1v1 + u2v2 + 4u3v3
is an inner product on R3. If it is not an inner product, list the axioms which do not hold.
Click here to see answer by Resolver123(6)

Question 1167650: Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A=
| 2 1 |
| -1 3 |
Click here to see answer by Resolver123(6)

Question 1167394: How does the rank of the following matrix depend on the value of t?
(1,1,t)
(1,t,1)
(t,1,1)

Click here to see answer by Resolver123(6)

Question 1160838: Find an orthonormal basis of the plane x−4y−z=0.

Click here to see answer by ikleyn(53419)

Question 1165856: Exercise 9 Determine whether the transformation T : R3 → R2 is a linear transformation.
Matrix2x1 T(x)=( x1, x1 + x2 +x3)
Click here to see answer by ikleyn(53419)

Question 1160376: pi =-2Q^(2)+9Q-4 and TC= 4-Q
What is the value of Fixed Cost?
What is the value of variable cost?

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Question 1160375: pi =-2Q^(2)+9Q-4 and TC= 4-Q
What is the value of the maximum profit?
Click here to see answer by ikleyn(53419)

Question 1160374: \pi =-2Q^(2)+9Q-4 and TC= 4-Q
For what values of Q does the firm break even
Click here to see answer by ikleyn(53419)

Question 1165009: The Cream and Custard Bakery makes both coffee cakes and Danish in large pans. The main
ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available
and the maximum demand for coffee cakes is 8. Five pounds of flour and 2 pounds of sugar
are required to make one pan of coffee cake, and 5 pounds of flour and 4 pounds of sugar are
required to make one pan of Danish. One pan of coffee cake has a profit of PhP 1, and one pan
of Danish has a profit of PhP 5. Determine the number of pans of cake and Danish that the
bakery must produce each day so that profit will be maximized.
Click here to see answer by ikleyn(53419)

Question 1168301: Fruit & Veg shop sells water in 5-litre bottles.
2.1 On Wednesday Fruit & Veg shop received $2 530 from selling 5-litre bottles of water at $11.50
per bottle. How many litres of water were sold on that day?
2.2 On Thursday, the shop received x $ by selling 5-litre bottles of water at 50 . 11 $ each. In terms
of ,x how many litres of water were sold on that day?
2.3 On Friday the shop received $(x  20) by selling 5-litre bottles of water at $9 each. In terms
of x, how many litres of water were sold on that day?
2.4 If the number of bottles sold on Thursday equal to the number of bottles sold on Friday, how many
bottles of water were sold in each of these two days?
Click here to see answer by CPhill(2138)

Question 1168385: A)Consider the vector space P2. Define the inner product,
⟨p, q⟩ = ∫(from0 to1) p(x)q(x) dx
Use the Gram-Schmidt process to transform the standard basis S = {1, x, x^2} into an orthonormal basis.

A.1)Express r(x) = 1 + x + 4x^2 as a linear combination of the vectors in the orthonormal basis for P2 found in the previous exercise.

Click here to see answer by CPhill(2138)

Question 1168921: Let vector p=2−x+x^2 and let B be the basis for P2:
{1+x,1+x^2,x+x^2}.
Find [p]B.
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Question 1170702: Let M=[{0,-2},{4,6}]. Find formulas for the entries of Mn, where n is a positive integer.
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Question 1174344: Q−4: [6+4 marks] Let S={v_1,v_2,v_3} be a linearly independent set of vectors in〖 R〗^n and T={u_1,u_2,u_3}, where u_1=v_1+v_2+v_3, u_2=v_2+v_3 and u_3=v_3.
a-Determine whether each of v_1,v_2 and v_3 is a linearly combination of vectors in T.
b-Show that T is linearly independent set.

Click here to see answer by CPhill(2138)

Question 1174495: Q−4: [6+4 marks] Let S={v_1,v_2,v_3} be a linearly independent set of vectors in〖 R〗^n and T={u_1,u_2,u_3}, where u_1=v_1+v_2+v_3, u_2=v_2+v_3 and u_3=v_3.
a- Determine whether each of v_1,v_2 and v_3 is a linearly combination of vectors in T.
b- Show that T is linearly independent set.

Click here to see answer by CPhill(2138)

Question 1175538: Q−4: [6+4 marks] Let S={v_1,v_2,v_3} be a linearly independent set of vectors in〖 R〗^n and T={u_1,u_2,u_3}, where u_1=v_1+v_2+v_3, u_2=v_2+v_3 and u_3=v_3.
A- Determine whether each of v_1,v_2 and v_3 is a linearly combination of vectors in T.
B- Show that T is linearly independent set.
Click here to see answer by CPhill(2138)

Question 1178699: this time, our immune system is the best defense . With this, a Melegail wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food A contains 2 units /kg of Vitamin A and 1 unit of /kg of vitamin C. Food B contains 1 unit/kg of vitamin A and 2 units/kg of Vitamin C. It costs 50.00 per kg to purchase food A and 70.00 per kg to purchase Food B. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.
Click here to see answer by CPhill(2138)

Question 1183383: Let A ∈ Mn×n(F), and let T : Mn×n(F) → Mn×n(F) be the linear transformation X → AXA, for X ∈
Mn×n(F). What is the rank of T? Show that there exists a matrix B such that ABA = A.

Click here to see answer by CPhill(2138)

Question 1185460: Table: Rate of the Cricket Chirps
Temperature in ℉
40
60
80
100
120
Rate
(Number of Chirps per Minute)
0
86
172
258
516

a.) Find a formula for g if g(t) represents the number of chirps per minute a cricket makes at temperature t degrees Fahrenheit.
b.) If f(c) represents the Fahrenheit reading that corresponds to a Celsius reading of c, which between the two functions g(f(t)) or f(g(t)) represents the number of chirps per minute a cricket makes when the temperature is c degrees Celsius?

c.) For the function in (b), write a formula for this and name it function h.

d.)Find the rate at which a cricket chirps if the temperature is __℉? __℃?
e.)Find the slope of the function g(t), h(c), and f(c). What does the slope of g(t) mean within the context of the problem?

Click here to see answer by CPhill(2138)

Question 1186205: The Intellectual Company produces a chemical solution used for cleaning carpets. This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y. Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds. Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound. Chemical A contains one unit of agent X and three units of agent Y. Chemical B contains seven units of agent X and one unit of agent Y.
a. Set up the following:
i. Variables
ii. Constraints
iii. Objective Function
b. Find the minimum cost
c. Determine the best combination of the ingredients to minimize the cost.

Click here to see answer by ikleyn(53419)

Question 1186248: The Intellectual Company produces a chemical solution used for cleaning carpets. This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y. Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds. Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound. Chemical A contains one unit of agent X and three units of agent Y. Chemical B contains seven units of agent X and one unit of agent Y.
a. Set up the following:
i. Variables
ii. Constraints
iii. Objective Function
b. Find the minimum cost
c. Determine the best combination of the ingredients to minimize the cost.
Click here to see answer by ikleyn(53419)

Question 1186463: To greet the 2020 SHS graduates, a tarpaulin is to be set along the national highway. If the area of the tarp is to be 35/4 m^2 and its perimeter 27/2 meters, what should be the dimensions of the tarpaulin

Click here to see answer by CPhill(2138)

Question 1191317: Determine whether the relation R on the set of all real numbers is reflective, symmetric, and/or transitive, where open parentheses x comma y close parentheses element of R if and only if x y greater or equal than 0. Is the relation R an equivalence relation?
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